MANUAL FOR TIDAL HEIGHTS

ANALYSIS AND PREDICTION

M.G.G. Foreman

by

Pacific Marine Science Report 77-10

INSTITUTE OF OCEAN SCIENCES, PATRICIA BAY

Sidney, B.C.

For additional copies or further information please either write to:

Department of Fisheries and Oceans

Institute of Ocean Sciences

P.O. Box 6000

Sidney, B.C. V8L 4B2

Canada

or see http://www.pac.dfo-mpo.gc.ca/sci/osap/projects/tidpack/tidpack e.htm

Pacific Marine Science Report 77-10

MANUAL FOR

TIDAL HEIGHTS ANALYSIS AND PREDICTION

by

M.G.G. Foreman

Institute of Ocean Sciences

Patricia Bay

Victoria, B.C.

1977

Revised September 1979

Reprinted May 1984

Revised November 1993

Revised July 1996

Revised October 2004

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CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 USE OF THE TIDAL HEIGHTS ANALYSIS

COMPUTER PROGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Routines Required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Data Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Program Conversion, Modifications, Storage and Dimension Guidelines . . . . . . 6

2 TIDAL HEIGHTS ANALYSIS PROGRAM DETAILS . . . . . . . . . . . . 8

2.1 Constituent Data Package . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Astronomical variables . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Choice of constituents and Rayleigh comparison pairs . . . . . . . . . . . 92.1.3 Satellite constituents and nodal modulation . . . . . . . . . . . . . . 152.1.4 Shallow water constituents . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Least Squares Method of Analysis . . . . . . . . . . . . . . . . . . . . 162.2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Solution of the matrix equation, the condition number and

statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Modifications to the Least Squares Analysis Results . . . . . . . . . . . . . 23

2.3.1 Astronomical argument and Greenwich phase lag . . . . . . . . . . . . 232.3.2 Nodal corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.3 Final amplitude and phase results . . . . . . . . . . . . . . . . . . . 272.3.4 Inferred constituents . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 USE OF THE TIDAL HEIGHTS PREDICTION

COMPUTER PROGRAM . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Routines Required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Data Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Program Conversion, Modifications, Storage and Dimension Guidelines . . . . . 32

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4 TIDAL HEIGHTS PREDICTION PROGRAM DETAILS . . . . . . . . . . 33

4.1 Problem Formulation and the Equally Spaced Predictions Method . . . . . . . 334.2 The High and Low Tide Prediction Method . . . . . . . . . . . . . . . . . 34

5 CONSISTENCY OF THE ANALYSIS AND PREDICTION

PROGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 APPENDICES

7.1 Standard Constituent Input Data for the Tidal HeightsAnalysis Computer Program . . . . . . . . . . . . . . . . . . . . . . . . 40

7.2 Sample Tidal Station Input Data for the Analysis Program . . . . . . . . . . 477.3 Final Analysis Results Arising from the Input Data of

Appendices 7.1 and 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.4 Sample Input for the Tidal Heights Prediction Program . . . . . . . . . . . . 517.5 Tidal Heights Prediction Results Arising from the Input Data of

Appendix 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

iii

PREFACE

This report is intended to serve as a user’s manual to G. Godin’s tidal heights analysisand predictions programs, revised along lines suggested by Godin. In addition to describinginput and output of these programs, the report gives an outline of the methods used; a fullpresentation of which can be found in Godin (1972) and Godin and Taylor (1973).

Users who wish to receive updates of these programs and manual should send their names,addresses, and type of computer used, to the author.

iv

ACKNOWLEDGEMENTS

The writer wishes to thank G. Godin for his guidance during the computer program revi-sions, J. Taylor and R.F. Henry for their helpful suggestions, and R. Rutka for transferring themanuscript to TEX.

1

1 USE OF THE TIDAL HEIGHTS ANALYSIS

COMPUTER PROGRAM

1.1 General Description

This program analyses the hourly height tide gauge data for a given period of time. Ampli-tudes and Greenwich phase lags are calculated via a least squares fit method coupled with nodalmodulation for only those constituents that can be resolved over the length of the record. Unlessspecified otherwise, a standard data package of 69 constituents will be considered for inclusionin the analysis. However, up to 77 additional shallow water constituents can be requested. Ifthe record length is such that certain important constituents are not included directly in theanalysis, provision is made for the inference of the amplitude and phase of these constituentsfrom others. Gaps within the tidal record are permitted.

1.2 Routines Required

(1) MAIN . . . . . . reads in some of data, controls most of the output and calls otherroutines.

(2) INPUT . . . . . . reads in the hourly height data for the desired time period andchecks for errors.

(3) UCON . . . . . . chooses the constituents to be included in the analysis via theRayleigh criterion

(4) SCFIT2 . . . . . . finds the least squares fit to an equally spaced time series usingsines and cosines of specified frequencies as fitting functions.

(5) VUF . . . . . . reads required information and calculates the nodal corrections forall constituents.

(6) INFER . . . . . . reads required information and calculates the amplitude and phaseof inferred constituents, as well as adjusting the amplitude andphase of the constituent used for the inference.

(7) CHLSKY . . . . . . solves the symmetric positive definite matrix equation resultingfrom a linear least squares fit.

(8) GDAY . . . . . . returns the consecutive day number from a specific origin for anygiven date and vice versa.

(9) ASTR . . . . . . calculates ephermides for the sun and moon.

(10) OUTPUT . . . . . . writes predicted hourly heights to the output file.

(11) SCULP . . . . . . scales up amplitudes to compensate for moving average filters.

2

1.3 Data Input

For a computer run of the tidal heights analysis program, two logical units are used fordata input. Logical unit number 8 contains the tidal constituent information while logical unit 4contains the hourly heights and information relating to the type of analysis and output required.A listing of the standard constituent information for logical unit 8 and a sample set of inputfor logical unit 4 are given in Appendices 7.1 and 7.2 respectively.

Logical unit 8 expects four types of data:

(i) One card each for all possible constituents, KONTAB, to be included in the analysis along withtheir frequencies, FREQ, in cycles/h and the constituent with which they should be comparedunder the Rayleigh criterion, KMPR. The format used is (4X,A5,3X,F13.10,4X,A5). UnlessKONTAB is specifically designated on logical unit 4 for inclusion, a blank data field for KMPR

results in the constituent not being included in the analysis.

A blank card terminates this data type.

(ii) Two cards specifying values for the astronomical arguments SO,HO,PO,ENPO,PPO,DS,DH,DP,

DNP,DPP in the format (5F13.10).

SO = mean longitude of the moon (cycles) at the reference time origin;HO = mean longitude of the sun (cycles) at the reference time origin;PO = mean longitude of the lunar perigee (cycles) at the reference time origin;

ENPO = negative of the mean longitude of the ascending node (cycles) at thereference time origin;

PPO = mean longitude of the solar perigee (perihelion) at the reference time origin.

DS,DH,DP,DNP,DPP are their respective rates of change over a 365-day period at the refer-ence time origin.

Although these argument values are not used by the program that was revised in October1992, in order to maintain consistency with earlier programs, they are still required as input.Polynomial approximations are now employed to more accurately evaluate the astronomicalarguments and their rates of change.

(iii) At least one card for all the main tidal constituents specifying their Doodson numbers andphase shifts along with as many cards as are necessary for the satellite constituents. Thefirst card for each such constituent is in the format (6X,A5,1X,6I3,F5.2,I4) and containsthe following information:

KON = constituent name;II,JJ,KK,LL,MM,NN = the six Doodson numbers for KON;

SEMI = the phase correction for KON;NJ = the number of satellite constituents.

A blank card terminates this data type.

If NJ>0, information on the satellite constituents follows, three satellites per card, inthe format (11X,3(3I3,F4.2,F7.4,1X,I1,1X)). For each satellite the values read are:

LDEL,MDEL,NDEL = the last three Doodson numbers of the main constituent

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subtracted from the last three Doodson numbers of thesatellite constituent;

PH = phase correction of the satellite constituent relative to the phaseof the main constituent;

EE = amplitude ratio of the satellite tidal potential to that of themain constituent;

IR = 1 if the amplitude ratio has to be multiplied by the latitudecorrection factor for diurnal constituents,

= 2 if the amplitude ratio has to be multiplied by the latitudecorrection factor for semidiurnal constituents,

= otherwise if no correction is required to the amplitude ratio.

(iv) One card specifying each of the shallow water constituents and the main constituents fromwhich they are derived. The format is (6X,A5,I1,2X,4(F5.2,A5,5X)) and the respectivevalues are:

KON = name of the shallow water constituent;NJ = number of main constituents from which it is derived;

COEF,KONCO = combination number and name of these main constituents.

The end of these shallow water constituents is denoted by a blank card.

Logical unit 4 contains six types of data:

(i) One card for the variables IOUT1,RAYOPT,ZOFF,ICHK,OBSFAC,INDPR,NSTRP in the format(I2,2X,F4.2,2X,F10.0,I2,3X,F10.7,215).

IOUT1 = 6 if the only output desired is a line printer listing of results,= 2 if both analysis output and listing are desired;

RAYOPT = Rayleigh criterion constant value if different from 1.0;ZOFF = constant to be subtracted from all the hourly heights;ICHK = 0 if the hourly height input data is to be checked for format errors,

= otherwise if this checking to be waived;OBSFAC = scaling factor, if different from 0.01, which will multiply the hourly

observations, in order to produce the desired units for the final constituentamplitudes. (e.g. if the hourly observations are in mm/s and the final unitsare to be ft/sec, then this variable would be set to 0.0032808.);

INDPR = 1 if hourly height predictions based on the analysis results are to becalculated and written onto device number 10. If there is inference, thisparameter value will also give the rms residual error after inferenceadjustments have been made,

= 0 if no such predictions are desired;NSTRP = number of successive moving average filters that have been applied to the

original data.

If NSTRP>0, then TIMINT and (LSTRP(I),I=1,NSTRP) will be read on a following card, inthe format (F10.0,1015), and suitable amplitude corrections will be applied to compensate forthe smoothing effect of these filters.

TIMINT = sampling interval, in minutes, of the original unfilteredrecord;

4

(LSTRP(J),J=1,NSTRP) = number of consecutive observations used in computing eachof the NSTRP moving average filters.

(ii) One card for each possible inference pair. The format is (2(4X,A5,E16.10),2F10.3) andthe respective values read are:

KONAN & SIGAN = name and frequency of the analysed constituent to be used for theinference;

KONIN & SIGIN = name and frequency of the inferred constituent;R = amplitude ratio of KONIN to KONAN;

ZETA = Greenwich phase lag of the inferred constituent subtracted from theGreenwich phase lag of the analysed constituent.

These are terminated by one blank card.

(iii) One card for each shallow water constituent, other than those in the standard 69 constituentdata package, to be considered for inclusion in the analysis. The Rayleigh comparisonconstituent is also required and the additional shallow water constituent must be found indata type (i) of logical unit 8, but have a blank data field where the Rayleigh comparisonconstituent is expected. The format is (6X,A5,4X,A5) and a blank card is required at theend.

(iv) One card in the format (I1,1X,10I2) specifying the following information on the periodof the analysis:

INDY = 8 indicates an analysis is desired for the upcoming period;= 0 indicates no further analyses are required;

IHH1,IDD1,IMM1,IYY1,ICC1 = hour, day, month, year and century of the beginning ofthe analysis (measured in time ITZONE of input data (v));

IHHL,IDDL,IMML,IYYL,ICCL = hour, day, month, year and century of the end of theanalysis.

If ICC1 or ICCL are zero, their value is reset to 19.

(v) One card in the format (I1,4X,A5,3A6,A4,A3,1X,2I2,I3,I2,5X,A5) containing the fol-lowing information on the tidal station:

INDIC = 1 if J card output is desired (no longer used),= otherwise if not;

KSTN = tidal station number;(NA(J),J=1,4) = tidal station name (22 characters maximum length);

ITZONE = time zone of the hourly observations;LAD,LAM = station latitude in degrees and minutes;LOD,LOM = station longitude in degrees and minutes;

IREF = reference station number.

(vi) The hourly height data cards contain the following information in the format (I1,1X,I5,4X,I2,1X,3I2,12A4).

KOLI = 1 or 2 indicates whether this specific card is the first or second

5

one for that day,= otherwise indicates a non-data card which is ignored;

JSTN = tidal station number;IC,ID,IM,IY = century, day, month and year of the heights on this card. If IC=0,

it is reset to 19;(KARD(J),J=1,12) = hourly heights in integer form. The final constituent amplitudes

unless a are in units 1/100 of those for the hourly height nonzerofor OBSFAC is read (see (i)). Missing values should be specified asa blank field or 9999.

When KOLI=1, the first hourly height on the data card is assumed to be at 0100 h andwhen KOLI=2, it is assumed to be at 1300 h. The time zone of these observations determinesthe nature of the Greenwich phase lag (see Section 2.3.1).

After the initial analysis of a computer run is completed, control returns to input (iv).Successive cards are read then until either a 0 or 8 value is found for INDY.

The hourly height data cards need not begin and end so as to include exactly the analysisperiod. The program ignores data outside this range. However if more than one analysis isdesired from a single job submission and hourly height data cards do extend beyond the firstanalysis period, care should be taken to ensure that one of these cards does not have KOLI=0

or blank, otherwise the job will be terminated. This is because all successive cards after theone containing the last hour of the desired analysis period are read in input (iv) format.

1.4 Output

Three logical units are used for the output of results from the tidal heights analysis program.Device number 6 is the line printer, 2 is used for analysis results and 10 contains hourlysynthesized values based on the analysis results; 6 is required for all program runs whereas theuse of 2 and 10 is controlled by the input variables IOUT1 and INDPR which are read fromdevice 4.

Recommendations for the use of moving average filters on the elevation data prior to submis-sion for analysis, and the scaling compensation method used in the improved analysis programare found in Foreman (1978) or Godin (1972).

When IOUT1 is 6, INDPR is other than 1, and there are no inferred constituents, the onlyoutput is two pages on the line printer. The first of these lists the constituents included inthe least squares fit, their frequencies in cycles/h (although eight decimal places are given,depending on computer accuracy, less than this number may be significant), the C and Scoefficient values (see Section 2.2.1) measured in units OBSFAC times those for the hourly heights,and their respective standard deviation estimates. It also specifies the number of hourly heightobservations (excluding gaps) within the analysis period, the average and standard deviation ofthe original observations, the root mean square residual error, and the matrix condition number.In the columns titled AL, GL, A, and G, the second page respectively lists the amplitudes andphases (degrees) obtained for each constituent from the C and S coefficient values, and thesame amplitudes and phases after nodal modulation and astronomical argument adjustments.The initial and final hour of the analysis are also specified along with the Rayleigh criterionconstant (‘separation’), the midpoint of the analysis period, the total number of possible hourlyobservations in the analysis period, and the total number of possible observations used in theanalysis. This last value includes gaps in the record and is the largest odd number less thanor equal to the total number of possible hourly observations (if the total number of possible

6

hourly observations is an even number, the last hour is ignored). If there is at least oneinferred constituent, page 2 results are repeated with the inclusion of inferred constituents andappropriate adjustments to the constituents from which the inferences were made. Appendix 7.3lists the final page of results obtained from the input value of Appendix 7.2.

The only effect of changing the value of IOUT1 to 2 (regardless of INDPR’s value) is tostore on file 2, the same information as the second (and third) page(s) of the line printer.The list of constituent names, amplitudes and Greenwich phase lags begins on line 5 of thisfile and is in the correct format for input to the tidal heights prediction program, namely(5X,A5,28X,F8.4,F7.2).

When INDPR equals 4, device 10 will contain hourly predictions calculated from the analysisresults. Values are specified only for the analysis period, including those intervals where therewere gaps in the original record, and are in the same measurement units and scaling as theoriginal data. The format used is the same as for input type (vi) of logical unit 4.

1.5 Program Conversion, Modifications, Storage and Dimension Guidelines

The source program and constituent data package described in this manual have beentested on various mainframe, PC and workstation computers at the Institute of Ocean Sciences,Patricia Bay. Although as much of the program as possible was written in basic FORTRAN,some changes may be required before the program and data package can be used on otherinstallations. Please write or call the author if any problems are encountered.

The program in its present form requires approximately 68,000 bytes for the storage of itsinstructions and arrays.

Changing the number or type of constituents in the standard data package may require somealterations to the analysis program. If constituents are added to the standard data package, thedimensions of several arrays may have to be altered. Restrictions on the minimum dimensionof such arrays are now given.

Let

MTOT be the total number of possible constituents contained in the data package(presently 146),

M be the number of constituents considered for inclusion in the analysis (presently69 plus the number of shallow water constituents specifically designated forinclusion),

MCON be the number of main constituents in the standard data package (presently 45),

MSAT be the sum of the total number of satellites for these main constituents and thenumber of main constituents with no satellites (presently 162 plus 8 for theversion of the constituent data package, listed in Appendix 7.1, that contains nothird-order satellites for both N2 and L2),

MSHAL be the sum for all shallow water constituents, of the number of main constituentsfrom which each is derived (presently 251).

Then in the main program, arrays KONTAB,FREQ and KMPR should have minimum dimensionMTOT+1; arrays KON,C,S,SIG,ERC,ERS,A,EPS,KO,AA and GD should have minimum dimensionM; array NKON should have dimension at least as large as the number of extra shallow waterconstituents specifically designated for analysis inclusion (its present maximum is 15); and arrays

7

Z and XP should be large enough to contain the hourly heights (and gaps) in the analysis period(its present maximum is 375 days).

In subroutine INPUT array Z should be dimensioned the same as in the main program,while KARD and IHT should be dimensioned 12.

In the other subroutine OUTPUT, Z is in a common block and should be dimensioned asin the main program, XP is in the argument list and need only have dimension 2, and arraysMONTH and IHT should have dimension 12 and 24 respectively.

In subroutine VUF, arrays VU and F should have minimum dimension MTOT; arrays KON

and NJ should have minimum dimension MTOT+1; arrays II,JJ,KK,LL,MM,NN and SEMI shouldhave minimum dimension MCON+1; arrays EE,LDEL,MDEL,NDEL,IR and PH should have minimumdimension MSAT; and KONCO, COEF should have minimum dimension MSHAL+4.

In subroutine INFER, arrays KONAN,KONIN,SIGAN,SIGIN,R and ZETA can presently accom-modate a maximum of nine inferred constituents.

In subroutine SCFIT2, arrays X,XP,C,S,ERC,ERS and F should have the same dimension asZ,XP,C,S,ERC,ERS and SIG in the main program and arrays RHS and A should have minimumdimension 2M-1 and M(2M-1) respectively. AC and AS should have the size of A and care shouldbe taken that through their equivalence relationships, neither AC and AS, nor RHSC and RHSS

overlap.

Finally, in subroutine CHLSKY, arrays A and F should have minimum dimensions M(2M-1)

and 2M-1 respectively.

8

2 TIDAL HEIGHTS ANALYSIS PROGRAM DETAILS

2.1 Constituent Data Package

2.1.1 Astronomical variables

The astronomical variables required by the tidal analysis program were used by Doodson(1921) in his development of the tidal potential. From them one can calculate the position ofthe sun or moon, and hence the tide generating forces, at any time. These varaibles are:

S(t) = mean longitude of the moon;H(t) = mean longitude of the sun;P (t) = mean longitude of the lunar perigee;

N ′(t) = negative of the longitude of the mean ascending node;P ′(t) = mean longitude of the solar perigee (perihelion).

For H, N ′ and P ′ these longitudes are measured along the ecliptic eastward from themean vernal equinox position at time t; while for S and P they are measured in the eclipticeastward from the mean vernal equinox position at time t to the mean ascending mode of thelunar orbit, and then along this orbit. Together with the rates of change of these variables,τ the local mean lunar time, and the Doodson numbers for each tidal constituent, one cancalculate the constituent frequencies, their astronomical argument phase angles, V , and theirnodal modulation phase, u, and amplitude, f , corrections.

The values of the astronomical variables and constituent frequencies in the program arecalculated using the power series expansion formulae given on pages 98 and 107 of the Ex-

planatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical

Almanac (1961). These formulae were derived from Newcomb’s Tables of the Sun and a revisionof Brown’s lunar theory (used in the development of his Tables of Motion of the Moon) so thatit is in accord with Newcomb’s.

(For those interested, even higher ordered approximations can be found in Astronomical

Formulae for Calculators by Jean Meeus.) In particular, the astronomical variables and fre-quencies are calculated at the central hour of the analysis period and in order to gain precisiont0, the reference time origin, is taken to be 0000 ET.1 This latter date, it was felt, would becloser to the analysis period of most records than the previous choice of 0000 ET January 1,1901, and hence would yield more accurate results via the linear approximation.

In keeping with the choice of reference time origin and astronomical variable specifications, tshould be measured in Ephemeris time. However, the correction from Universal time is irregularand in most cases small, so it has been assumed for computational purposes that all observationsare recorded in ET.

2.1.2 Choice of constituents and Rayleigh comparison pairs

There is a maximum of 146 possible tidal constituents that can be included in the tidalanalysis, 45 of these are astronomical in origin (main constituents) while the remaining 101 are

1 Ephemeris Time (ET) is the uniform measure of time defined by the laws of dynamics and determined

in principle from the orbital motion of the Earth as represented by Newcomb’s Tables of the Sun. Universal

or Greenwich Mean Time is defined by the rotational motion of the Earth and is not rigorously uniform.

9

shallow water constituents.2 Because computation time (and cost) of the computer programincreases approximately as the square of the number of constituents included in the analysis,and because for many tidal stations, most of the shallow water constituents are insignificant, asmaller standard package was seen as adequate for general use. Based on the suggestions of G.Godin, it was decided that this package contain all the main constituents and 24 of the shallowwater. However, provision was made so that other shallow water constituents among the 77remaining could be included if desired.

The Rayleigh comparison constituent is used for the purpose of deciding whether or nota specific constituent should be included in the analysis. If F0 is the frequency of such aconstituent, F1 is the frequency of its Rayleigh comparison constituent and T is the time spanof the proposed record to be analysed, then the constituent will be included in the analysisonly if |F0 − F1|T ≥ RAY . RAY is commonly given the value 1 although it can be specifieddifferently in the program.

In order to determine the set of Rayleigh comparison pairs, it is important to consider,within a given constituent group (e.g. diurnal or semidiurnal), the order of constituent inclusionin the analysis as T (the time span of the record to be analysed) increases. Assuming this pointof view, the specific objectives used when constructing the set listed in Appendix 7.1 were:

(i) within each constituent group, when possible, have the order of constituent selection corre-spond with decreasing magnitude of tidal potential amplitude (as calculated by Cartwrightand Edden (1973)),

(ii) when possible, compare a candidate constituent with whichever of the neighbouring, alreadyselected constituents, that is nearest in frequency,

(iii) when there are two neighbouring constituents of relatively equal tidal potential amplitude,rather than waiting until the record length is sufficient to permit the selection of both atthe same time (i.e. by comparing them to each other), choose a representative of the pairwhose inclusion will be as early as possible. This will give information sooner about thatfrequency range, and via inference, still enable some information to be obtained on bothconstituents.

The Rayleigh comparison pairs chosen for the low frequency, diurnal, semidiurnal and ter-diurnal constituent groups are given in Tables 1, 2, 3 and 4 respectively. Figures given for thelength of record required for constituent inclusion assume a Rayleigh criterion constant value(input variable RAYOPT) of 1.0.

2Q1 and SIG1 provide an example of objective (iii). Because 2Q1 has a greater frequencyseparation for Q1 and hence would appear in an analysis of shorter record length than SIG1,it was chosen as the representative.

However, it can be seen in several cases, that it was not possible or feasible to adhereto all the objectives just outlined. Choosing a Rayleigh comparison constituent from the listof those constituents already included in the analysis proved to be difficult near the frequencyedges of constituent groups. Upward arrows indicate failure to uphold this objective. OO1 issuch a case. For it, the potential comparison pairs were SO1, K1 and J1. The first of thesewould result in both SO1 and OO1 appearing at the same later time than had J1 or K1 been

2 The criterion for selecting these main constituents was to include all the diurnal and semidiurnal

constituents with Cartwright and Edden (1973) tidal potential amplitudes greater than 0.00250, along with

M3 and the most important low frequency constituents. Section 2.1.3 gives the analogous shallow water

constituent criterion.

10

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5 6 4 �

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: 8 ;

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* : = < 5 , * 5 7 8 ; : ,

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D ' 2 , / E 8 + ;

D ' 0 1 5 / ' E - , /

C , ( 7 * 5 * 1 ' ( *

F ( 2 9 1 7 5 , ( G & H I J K I L F M I K I D N O I O I P G Q I R S P I B O I C N F I H F I H I L I T I H L F I H N F I P N S I U I L O I O O I Q H L I

J V

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X J [

8 X \ ] V ;O I

] W Z

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] ] J

8 W W ; 8 \ J Z \ ; 8 J ] V ; 8 J ] V ; 8 X Z Z ;

J K I K I B O I U I Q H L I

\ ] V

8 J \ [ ;G & H I

V X [ X

8 V X ; 8 J \ [ ; 8 Z \ W V X ; 8 \ W W ;

P G Q I R S P I H I H N F I L O I

V V J

8 Z Z W J ; 8 Z X \ Z ; 8 W ] \ ; 8 W ] \ ;

L F M I D N O I C N F I P N S I

[ \ ] \

8 Z Y J [ ; 8 V Z ] ; 8 V J J ;

H F I L I H L F I

1.3

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D ' 2 , / E 8 + ;

D ' 0 1 5 / ' E - , /

C , ( 7 * 5 * 1 ' ( *

F ( 2 9 1 7 5 , ( G H I J K L I M B I N O I B I B O I P Q N I R S N I R I N T L I & 4 Q I & I U I L I D I T I N L B I J U Q I

V W

8 X Y Z Y X ;N I

W [ [

8 \ M M \ Z ;L I

] ] M

8 V W Z ] ; 8 ] \ W ;

B I J U Q I

] \

8 ] V ; 8 M ] ; 8 M [ X ;

J K L I N O I & I

\ W Z W

8 V V \ X Z ;

N T L I T I N L B I

\ X \ M

8 M [ X ; 8 M W Y V ; 8 W W Y M ; 8 ] Y ;

G H I M B I B O I & 4 Q I

Z ]

8 W V W ; 8 M ; 8 M \ ] ; 8 W [ [ ;

R S R I U I D I

V V W M ]

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13

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14

chosen. Hence, information about OO1 would be unnecessarily delayed. Although, due to thetidal potential amplitude of J1, objective (i) is violated with both the second and third choices,it was felt that the third was a better compromise. With it, OO1 only appears 11 h soonerthan J1.

K2 is an example of an unavoidable violation of objective (i). Because it is so close infrequency to S2, its importance as a major semidiurnal constituent does not insure it an earlyinclusion in the analysis package.

Because shallow water constituents do not have a tidal potential amplitude, objective (i)does not apply to them. However, based on his experience, Godin was able to suggest ahierarchy of their relative importance. A further criteria used when selecting comparison pairsfor them was that no shallow water constituent should appear in an analysis before all the mainconstituents, from which it is derived, have also been selected. Table 5 shows that this has

Table 5 Shallow Water Constituents in the Standard Data Package.

Shallow Record Length (h) Component Main Constituents and RecordWater Required for Lengths (h) Required for Their Inclusion

Constituent Constituent Inclusion in the Analysis

SO1 4383 S2 355 O1 328

MKS2 4383 M2 13 K2 4383 S2 356

MSN2 4383 M2 13 S2 355 N2 662

MO3 656 M2 13 O1 328

SO3 4383 S2 355 O1 328

MK3 656 M2 13 K1 24

SK3 355 S2 355 K1 24

MN4 662 M2 13 N2 662

M4 25 M2 13

SN4 764 S2 355 N2 662

MS4 355 M2 13 S2 355

MK4 4383 M2 13 K2 4383

S4 355 S2 355

SK4 4383 S2 355 K2 4383

2MK5 24 M2 13 K1 24

2SK5 178 S2 355 K1 24

2MN6 662 M2 13 N2 662

M6 26 M2 13

2MS6 355 M2 13 S2 355

2MK6 4383 M2 13 K2 4383

2SM6 355 S2 355 M2 13

MSK6 4383 M2 13 S2 355 K2 4383

3MK7 24 M2 13 K1 24

M8 26 M2 13

15

been upheld for all shallow water constituents in the standard 69 constituent data package.

We recommend that the objectives outlined here be employed when choosing the Rayleighcomparison constituent for any additions to the list of possible constituents to be included inthe analysis.

2.1.3 Satellite constituents and nodal modulation

Doodson’s (1921) development of the tidal potential contains a very large number of con-stituents. Due to the great length of record required for their separation, several of these can beconsidered, for all intents and purposes, unanalysable. The standard approach to this problemis to form clusters consisting of all constituents with the same first three Doodson numbers.The major contributor in terms of tidal potential amplitude lends its name to the cluster andthe lesser constituents are called satellites.

The method of analysis uses this main and satellite constituent approach in the followingmanner. The Rayleigh criteria is applied to the main constituent frequencies to determinewhether or not they are to be included in the analysis. For each of those so chosen, we analyseat its frequency and obtain an apparent amplitude and phase. However, because these resultsare actually due to the cumulative effect of all the constituents in that cluster, an adjustmentis made so that only the contribution due to the main constituent is found. This adjustmentis called the nodal modulation.

In order to make the nodal modulation correction to the amplitude and phase of a mainconstituent, it is necessary to know the relative amplitudes and phases of the satellites. Asis commonly done, it is assumed in this program that the same relationship as is found withthe equilibrium tide (tidal potential), holds with the actual tide. That is, the tidal potentialamplitude ratio of a satellite to its main constituent is assumed to be equal to the correspondingtidal heights amplitude ratio, and the difference in tidal potential phase equals the difference intidal height phase.

The source of the tidal potential amplitude ratio, as found in the constituent data packageof Appendix 7.1, is Cartwright and Tayler (1971) and Cartwright and Edden (1973). Usingnew computation methods and the latest values for the astronomical constants, they obtainedmore accurate results than those from the previously used Doodson computations. It should benoted that in several cases (whenever the satellite arises via the third-order term), this versionof the constituent data package requires that the amplitude ratio be multiplied by a latitudecorrection factor.

Phase differences between satellites and main constituents arise when the tidal potentialdevelopment yields different trigonometric terms for these constituents. The common conventionis to express all terms in cosine form and so an extra − 1

4cycle phase shift is introduced if the

term was originally a sine. Satellites requiring such a shift are called third order. A further 1

2

cycle change is also introduced when all negative amplitudes are made positive.

Because several test analyses indicate less consistent results when third-order satellites areincluded in the N2 and L2 nodal modulation, Godin has decided to delete these from the presentstandard constituent data package. Instead he suggests that the results of analyses with thispackage should be compared with those of previous analyses in order to find the most suitableadjustment for these constituents.

The only other main constituents that do not have all their satellites included for nodalmodulation are the slow frequency constituents. For them, no satellites are specified. Becauselow frequency noise may be as much as an order of magnitude greater than the satellite con-

16

tributions, and Mm, Msf and Mf when they are detectable are often of shallow water origin,the effect of making corrections for the expected satellites would be to obscure further, ratherthan clarify the actual low frequency periodic signal.

Section 2.3.2 gives further details on the nodal modulation correction.

2.1.4 Shallow water constituents

Shallow water tidal constituents arise from the distortion of main constituent tidal oscilla-tions in shallow water. Because the speed of propagation of a progressive wave is approximatelyproportional to the square root of the depth of water in which it is travelling, shallow waterhas the effect of retarding the trough of a wave more than the crest. This distorts the originalsinusoidal wave shape and introduces harmonic signals that are not predicted in tidal potentialdevelopment. The frequencies of these derived harmonics can be found by calculating the effectof non-linear terms in the hydrodynamic equations of motion on a signal due to one or moremain constituents (see Godin (1972), pp. 154–164 for further details).

The shallow water constituents chosen for inclusion in the standard 69 constituent datapackage were suggested by G. Godin. They are listed in Table 5 and are derived only fromthe largest main constituents, namely M2, S2, N2, K2, K1 and O1, using the lowest types ofpossible interaction. The 77 additional shallow water constituents that can be included in theanalysis if so desired are derived from lesser main constituents and higher types of interaction.In the constituent data package listing of Appendix 7.1, they can be spotted by their lack of aRayleigh comparison constituent.

When shallow water effects are noticeable, main constituents, if they are close in frequency,may coexist or be masked by constituents of non-linear origin. The resultant nodal modulationwill be due to the pair and thus will not coincide to the calculated modulation of the mainconstituent. In suspected cases, the effectiveness of nodal corrections in a series of successiveanalyses will indicate the presence of pairs or emphasize the predominance of one constituentover the other. Table 6 (taken from unpublished notes of Godin) lists compound constituentswhich may coexist with or mask constituents of direct astronomical origin. In all cases exceptSO1 and MO3, the main rather than the compound constituent is included in the standardconstituent data package.

2.2 The Least Squares Method of Analysis

2.2.1 Formulation of the problem

The first stage in the actual analysis of tidal records is the least squares fit for constituentamplitude and phase. If the tidal record is of minimum length 13 h, the present programand data package insure that the constant constituents Z0 and M2 are always included in theanalysis. If σj for j = 1, M are the frequencies (cycles/h) of the other tidal constituentschosen for inclusion in the analysis by the Rayleigh criterion, then the problem is to find theamplitudes, Aj , and phases, φj , of the function C0 +

∑Mj=1

Aj cos[2π(σjti −φj)] that best fit the

series of observations y(ti), i = 1, N .3 Assuming N > 2M + 1 we see that it is impossible to

3 In order to minimize the loss of accuracy due to round off, the average of the hourly heights observations

is subtracted from all original values. The y(ti) values mentioned in all computations henceforth are actually

the resultant deviations. At the end of all calculations, C0 is adjusted by this mean value.

17

Table 6 Shallow Water Constituents that May Mask Main Constituents.

Main Constituent Component Constituent

which May Coexist at or

Near its Frequency

Q1 NK1

O1 NK1**

TAU1 MP1**

NO1* NO1**

P1 SK1**

K1 MO1

J1 MQ1

SO1 SO1

OQ2 OQ2**

EPS2 MNS2

2N2 O2**

MU2 2MS2

N2 KQ2**

GAM2 OP2**

M2 KO2**

L2 2MN2**

S2 KP2

K2 K2

MO3 MO3**

M3 NK3**

∗ With M1 as a satellite.

∗∗ The modulation or frequency of the compound constituent

is sufficiently different that the pair could be separated if a

long enough record of high precision were available.

solve the system y(ti) = C0 +∑M

j=1Aj cos[2π(σjti − φj)] exactly because it is overdetermined.

Hence, it is necessary to adopt a criterion which will enable unique optimum values for theparameters Aj and φj to be found. The most common optimization criterion used, and the onechosen here, is the least squares technique.

Re-expressing∑M

j=1Aj cos [2π(σjti − φj)] as

M∑

j=1

[Cj cos(2πσjti) + Sj sin(2πσjti)] ,

where Aj = (C2j + S2

j )1/2 and 2πφj = arctan Sj/Cj , so that the fitting function is linear in theparameters Sj and Cj and hence more easily solved, and rewriting y(ti) as yi, the objective ofthe least squares technique is to minimize

T =

M∑

i=1

[

yi − C0 −M∑

j=1

(Cj cos 2πσjti + Sj sin 2πσjti)

]2

,

18

Figure 1 The matrix equation Bx = y resulting from the least squares fit for constituentamplitudes and phases.

for C0 and all Cj , Sj j = 1,M . This is done by solving the following 2M + 1 simultaneous

equations for j = 1,M :

0 =∂T

∂C0

= 2N

∑

i=1

(

y1 − C0 −M∑

j=1

Cj cos 2πσjti −M∑

j=1

Sj sin 2πσjti

)

(−1);

0 =∂T

∂C0

= 2

N∑

i=1

(

y1 − C0 −M∑

j=1

Cj cos 2πσjti −M∑

j=1

Sj sin 2πσjti

)

(− cos 2πσjti);

0 =∂T

∂C0

= 2N

∑

i=1

(

y1 − C0 −M∑

j=1

Cj cos 2πσjti −M∑

j=1

Sj sin 2πσjti

)

(− sin 2πσjti).

This results in the matrix equation Bx = y of Figure 1.

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!!!!!!!!!!!!!!!!!!"

19

Gaps in the data record (i.e. missing hourly observations) are easily handled by the leastsquares method because it is not necessary that the observation times, ti, for i = 1, N be evenlyspaced. For example, if the analysis covers the total time period of 100 h but hours 50 to 74inclusive are missing, then t50 will correspond to the seventy-fifth hour. However, because thefollowing identities which simplify the summations require that the observation times be evenlyspaced, it is necessary that each of the matrix terms be calculated as the sum of contributionsover the data periods that contain no gaps. Assuming that [n0, n1] is the hour range of a sectionof record containing no gaps, we can substitute tk = k in the matrix coefficients expressionssince the times are at successive hours.

Using the relationships

cos a cos b = 1

2[cos(a + b) + cos(a − b)]

sin a sin b = 1

2[cos(a − b) − cos(a + b)]

sin a cos b = 1

2[sin(a + b) + sin(a − b)],

the formula for the sum of a geometric series, namely

a + ar+, . . . ,+arn = a(rn+1 − 1)

(r − 1),

and expressing cos x and sin x as the real and imaginary parts of eix, we obtain the identities:

n1∑

k=n0

cos kx =sin{[(n1 − n0 + 1)x]/2} cos{[(n1 + n0)x]/2}

sin(x/2),

andn1∑

k=n0

sin kx =sin{[(n1 − n0 + 1)x]/2} sin{[(n1 + n0)x]/2}

sin(x/2).

Hence the summation expressions in the least squares matrix can be simplified (with regard to

computer execution time) as follows.

n1∑

k=n0

cos(2πσ1k) cos(2πσ2k) =1

2

n1∑

k=n0

{cos[2πk(σ1 + σ2)] + cos[2πk(σ1 − σ2)]}

=1

2

(

sin[(n1 − n0 + 1)π(σ1 + σ2)] cos[(n1 + n0)π(σ1 + σ2)]

sinπ(σ1 + σ2)

+sin[(n1 − n0 + 1)π(σ1 − σ2)] cos[(n1 + n0)π(σ1 − σ2)]

sin π(σ1 − σ2)

)

,

n1∑

k=n0

sin(2πσ1k) sin(2πσ2k) =1

2

n1∑

k=n0

{cos[2πk(σ1 − σ2)] − cos[2πk(σ1 + σ2)]}

=1

2

(

sin[(n1 − n0 + 1)π(σ1 − σ2)] cos[(n1 + n0)π(σ1 − σ2)]

sinπ(σ1 − σ2)

− sin[(n1 − n0 + 1)π(σ1 + σ2)] cos[(n1 + n0)π(σ1 + σ2)]

sin π(σ1 + σ2)

)

,

20

n1∑

k=n0

sin(2πσ1k) cos(2πσ2k) =1

2

n1∑

k=n0

{sin[2πk(σ1 + σ2)] + sin[2πk(σ1 − σ2)]}

=1

2

(

sin[(n1 − n0 + 1)π(σ1 + σ2)] sin[(n1 + n0)π(σ1 + σ2)]

sin π(σ1 + σ2)

+sin[(n1 − n0 + 1)π(σ1 − σ2)] sin[(n1 + n0)π(σ1 − σ2)]

sin π(σ1 − σ2)

)

.

With these substitutions made in Figure 1, we have the least squares matrix equation Bx = y

generated in subroutine SCFIT2. Because B is symmetric it is sufficient to store only its uppertriangle consisting of 2M2 +3M +1 elements instead of the entire matrix of (2M +1)2 elements.

Partitioning the matrix equation Bx = y into the form

(

B11 B12

B21 B22

)(

c

s

)

=

(

yc

ys

)

,

where B11, B12, B21, B22, c, s, yc, ys have dimensions (M + 1) × (M + 1), (M + 1) × M ,M × (M +1), M ×M , (M +1)×1, M ×1, (M +1)×1, M ×1 respectively, it is easily seen whenn0 = −n1 that B12 and B21 become zero matrices and two smaller matrix equations, B11c = yc

and B22s = ys, result. The combined computation time to solve these equations is less thanthat of the original (see Section 2.2.2) so it is desirable to attain this condition when possible.Since the time origin of the hourly observations is arbitrary provided it is consistent with thatof the astronomical argument V , we can attain the desired condition for a record with no gapsby choosing the central hour of the record as the origin. (This requires that the total numberof observations be odd and is satisfied by ignoring the last observation, if the total is even.)Although there is generally no corresponding matrix simplification in the case of a record withgaps, for consistency with the foregoing choice, it is convenient to choose the central hour ofthe record universally as the time origin.

2.2.2 Solution of the matrix equation, the condition number and

statistical properties

Most of the discussion and development of the Cholesky factorization algorithm introducedin this section is taken directly from Forsythe and Moler (1967). Although all results anddiscussion are now stated only for the matrix B and the equation Bx = y, they apply as wellfor the partitioned systems B11, B11c = Yc and B22, B22s = Ys.

In addition to symmetry, a useful property of matrix B is its positive definiteness. Thisproperty requires that for all (2M + 1) × 1 dimensional vectors x 6= 0, xT Bx > 0.

The positive definiteness of B can be demonstrated by considering the overdetermined ma-trix equation y = Ax+e resulting from the system of equations y(ti) = C0+

∑Mj=1

(Cj cos 2πσjti+

Sj sin 2πσjti) + ei for i = 1, N where the vector xT = (C0, C1, C2, . . . , CM , S1, S2, . . . , SM ),yT = [y(t1), . . . , y(tN )] and e is the vector of residuals. It is easily seen that AT A = B,and so for any x 6= 0,

xT Bx = xT AT Ax = zT z =

N∑

i=1

z2i ,

where xT AT = zT = (z1, . . . , zN ).

21

It is worth mentioning that the overdetermined system y = Ax + e can be solved in manyways, depending on the criterion chosen for minimizing e. For our purposes, those methodswhich solve the system without changing the form of the matrix are impractical from a storage,processing time and rounding error point of view because the first dimension of A (= thenumber of hourly observations) is commonly 9000. However, minimizing eT e is equivalent tothe least squares criterion adopted here.

An important result for any positive definite symmetric matrix B is that it can be uniquelydecomposed in the form B = GGT , where G is a lower triangular matrix with positive diagonalelements.4 Expanding this relationship leads to the matrix element equalities:

bjj =

j∑

k=1

g2jk,

bij =

j∑

k=1

gikgjk for all i > j.

The algorithm resulting from using these equations in the proper order to find the elementsof G is known as Cholesky’s square root method for factoring a positive definite matrix (alsoattributed to Banachiewicz; see Faddeev and Faddeeva (1963)). Unlike other matrix decom-position methods such as Gaussian elimination, it does not have to search for, and divide bypivots. Such techniques must insure that the reduced matrix elements are not too large so thatrounding errors and loss of accuracy do not occur. In Cholesky’s method however, we can seethat |gij | ≤

√bii for all i, j and so upper bounds for the elements of G always exist.

Once B has been decomposed into the upper and lower triangular matrices, it is a relativelyeasy matter to solve the matrix solution. This is done by breaking down the equation GGT x = y

into Gb = y and GT x = b. Because of the triangular nature of G, these equations can be solvedby forward and backward substitution for b and x respectively.

The amount of arithmetic in a matrix algorithm is usually measured by the number ofmultiplicative operations (i.e. multiplications and divisions) used, since there are normally ap-proximately the same number of additive operations. For a matrix of dimension n × n, theCholesky factorization algorithm requires n square roots and approximately 1

6n3 multiplica-

tions. This compares favourably with the 1

3n3 multiplications required by Gaussian elimination

(Wilkinson, 1967) to produce a triangular matrix.

Wilkinson (1967) suggests a factorization of B into LDLT , where L is a lower triangularmatrix and D is a positive diagonal matrix, that involves no more multiplications than Choleskyand avoids the square roots. However, assuming that the time ratio of a square root operationto a multiplication is 15:1 (approximate ratio for the IBM 370-168) and that all 69 constituentsin the data package are included in the analysis (i.e. n = 137) the time saved by eliminating thesquare roots in only 0.5%. Furthermore, some of this gain would be replaced by time requiredfor storing and retrieving information from the additional matrix D, and for the n additionaldivision operations each time a solution is calculated by forward and backward substitution.Hence the factorization was not adopted in the present program.

Because the time required for the factorization of B varies as the cube of the number ofunknowns, an approximate four-fold time reduction should result when the tidal record has no

4 If B is symmetric but not positive definite a similar decomposition exists. However, some elements of

G may be complex or, in the degenerate case, zero along the diagonal.

22

gaps and the partitioned rather than the original matrix equations are solved. However, as thefollowing table of execution times for sections of subroutine SCFIT2 demonstrates, significantimprovements can also be expected in the time required for matrix generation, and error cal-culation. The values shown in Table 7 were obtained on an IBM 370-168 computer with a34-constituent analysis of a 38-day tidal record.

A rough indication of the round-off difficulties associated with solving the equation Bx = yis given by the matrix condition number. Although several different definitions for a conditionnumber exist, an appropriate one for our purposes, in the sense that it pertains to least squaresmatrices and is easily calculated, is specified by Davis and Rabinowitz (1961). Its developmentis as follows.

Table 7 Comparison of Processing Times between the Partitioned

and Non-Partitioned Matrix Equation Solutions.

Components of Matrix Partitioned Matrix Non-Partitioned

Solution System Times (s) Matrix System (s)

Parameter initializations

and right-hand generation 0.347 0.346

Matrix generation 0.059 0.178

Matrix factorization 0.049 0.146

Solution 0.010 0.018

Error calculation 0.128 0.403

If {b1, . . . bn} are n-dimensional vectors such that the matrix

B =

b1

...bn

(b1 . . . bn)

=

b1 · b1 . . . b1 · bn...

...b1 · b1 . . . b1 · bn

,

then it can be shown that 0 ≤ det(B) ≤ ‖b1‖ ‖b2‖, . . . , ‖bn‖ where if bj = (bj1, . . . , bjn), thenorm ‖bj‖ = (

∑ni=1

b2ji)

1/2. Furthermore, det(B) = 0 if and only if the vectors are linearlydependent, and det(B) = ‖b1‖, . . . , ‖bn‖ if and only if they are orthogonal (i.e. bi · bj = 0 fori 6= j). This determinant is known as the Gram determinant of the system {b1, . . . ,bn} and isthe square of the n-dimensional volume of the parallelepiped whose edges are these vectors.

Since it can be shown that all least squares matrices can be expressed in this manner, thisresult can be applied to our situation. In particular when the vectors are normalized so that‖bi‖ = 1, the actual value of det(B) will always be bounded and provide a measure of thelinear independence of the system, and hence round-off difficulties encountered in solving theequation. A value close to 1 will mean near orthogonality, a virtually diagonal matrix for B,and thus an easy solution. On the other hand, a value close to 0 will mean that at least tworows are near scalar multiples of one another, and thus greater accuracy problems will occurwhen their difference is calculated during the equation solution.

For our particular case observe that det(B) = det(GGT ) = (det G)2 =∏n

i=1g2

ii, and that Bcan be written as

GGT =

g1 · g1 . . . g1 · gn...

...gn · gn . . . gn · gn

,

23

where

GT =

g11 g21 . . . gn1

0 g22 . . . gn2

...0 . . . 0 gnn

= (g1,g2 . . . gn).

. . .. .. . . . . ..........

Since bjj =∑j

k=1g2

jk, ‖gj‖ =√

bjj and the determinant of the matrix resulting from

normalizing the gj vectors is∏n

i=1(g2

ii/bii). The square root of this value is the volume of

the n-dimensional parallelepiped whose edges are these normalized vectors and is the quantitycalculated as the condition number of the matrix B.

The statistical properties of the least squares fit solution can be found in any analysis ofvariance or regression model text. They are outlined briefly as follows.

Reverting to the overdetermined problem statement, the least squares objective can bestated as finding the vector x in y = Ax + e such that eT e is minimized. This yields thesolution x = (AT A)−1AT y.

The total sum of squares is yT y and the sum of squares due to regression is xT AT y.Their difference is the residual error sum of squares and this difference divided by the degreesof freedom in the fit is the residual mean square error (MSE). “Degrees of freedom” is thedifference between the number of hourly observations (excluding gaps) and A the number ofparameters fit in the analysis. If there were M constituents including Z0 chosen for the analysis,the degrees of freedom would be N − 2M + 1.

If it is assumed, as is commonly done, that the vector e is distributed normally with 0standard deviation and σ2I variance, where I is the unit diagonal matrix, then the varianceof x is (AT A)−1σ2. Since the mean square residual error is an unbiased estimator for σ2, anestimate of the standard deviation of xi, the ith element of x, is

√

(µµµTi (AT A)−1µµµi)MSE ,

where µµµi is the vector with one in the ith position of zeros elsewhere.

2.3 Modifications to the Least Squares Analysis Results

2.3.1 Astronomical argument and Greenwich phase lag

Instead of regarding each tidal constituent as the result of some particular component ofthe tidal potential, an artificial causal agent can be attributed to each constituent in the formof a fictitious star which travels around the equator with an angular speed equal to that ofits corresponding constituent. Making use of this conceptual aid, the astronomical argument,V (L, t), of a tidal constituent can then be viewed as the angular position of this fictitiousstar relative to longitude, L, and at time, t. Although the longitudinal dependence is easilycalculated, for historical reasons L is generally assumed to be the Greenwich meridian, and Vis reduced to a function of one variable.

The Greenwich phase lag, g, is the difference between this astronomical argument for Green-wich and the phase of the observed constituent signal. Its value is dependent upon the timezone in which the hourly heights of the record were taken. This means that when phases atvarious stations, not necessarily in the same time zone, are compared, they must be reduced to

24

a common zone in order to avoid spurious differences due to difference relative times. Specif-ically, if σ is the constituent frequency and g(j + ∆j) and g(j) are the Greenwich phase lagsevaluated for time zones j + ∆j and j respectively (e.g. Pacific Standard Time is +8), then

g(j + ∆j) = g(j) − (∆j)σ.

Although these adjustments are easily calculated, they can be tedious because each constituentmust be handled individually. Therefore, to avoid possible misinterpretation of phases fromnearby stations of subsequent phase alterations, it is suggested that all observations be recordedin, or converted to, GMT.

The calculation of g (see Section 2.3.3) requires that the astronomical argument need onlybe evaluated at one time, the central hour of the analysis period. For a particular main con-stituent, it is calculated as

V = i0τ + j0S + k0H + l0P + m0N′ + n0P

′,

where i0, j0, k0, l0,m0, n0 are the Doodson numbers of the constituent and S,H,P,N ′, P ′ arethe astronomical variables defined in Section 2.1.1. The variable, τ , the number of mean lunardays from an absolute time origin is calculated as sum of the local mean solar time from thisorigin and (H − S), and so need not be read from the data cards.

For shallow water constituents, the astronomical argument is calculated as the linear com-bination of the coefficient number and the astronomical argument of the main constituents fromwhich it is derived. For example,

VMSN2= VM2

+ VS2− VN2

and V2MK5= 2VM2

+ VK1.

2.3.2 Nodal corrections

Most of this section has been taken from the unpublished notes of G. Godin which werewritten subsequent to the Cartwright and Tayler (1971) and Cartwright and Edden’s (1973)recalculation of the tide-generating potential. The material presented here is intended to givegreater detail than that of Section 2.1.3.

Due to the presence of satellites in a given cluster, it is known from tidal potential theorythat the analysed signal found at the frequency, σj , of the main constituent is actually theresult of

aj sin(Vj − gj) +∑

k

Ajkajk sin(Vjk − gjk) +∑

l

Ajlajl cos(Vjl − gjl)

for the diurnal and terdiurnal constituents of direct gravitational origin, and

aj cos(Vj − gj) +∑

k

Ajkajk cos(Vjk − gjk) +∑

l

Ajlajl sin(Vjl − gjl)

for the slow and semidiurnal constituents. The variables, a, g and V , are the true amplitude,Greenwich phase and astronomical argument, respectively, at the central time of the recordfor all the constituents. Single j subscripts refer to the major contributor while jk and jlsubscripts refer to satellites originating from tidal potential terms of the second and third orderrespectively. A is the element of the interaction matrix resulting from the interference of asatellite with the main constituent.

25

It is the convention in tides and an assumption for our least squares fit that all constituentsarise through a cosine term and positive amplitude, i.e. the contribution for a constituent whoseastronomical argument is Vj and whose Greenwich phase is gj , is expected to be in the formaj cos(Vj − gj) for aj > 0. However, the diurnal and terdiurnal constituents, assuming that theyare due to second order terms in the tidal potential, actually arise through a bj sin(Vj − gj)term where bj may be negative. Hence a phase correction (variable SEMI read in data input(iii) from logical unit 8) of either − 1

4or − 3

4cycles is necessary, i.e.

bj sin(Vj − gj) = |bj | cos(

Vj − gj − 1

4

)

bj ≥ 0,

= |bj | cos(

Vj − gj − 3

4

)

bj < 0.

Similarly, an adjustment of 1

2cycle will only be necessary for slow and semidiurnal main con-

stituents if the tidal potential amplitude is negative.

Making these changes, the combined result of a constituent cluster in the diurnal andterdiurnal cases is

|aj | cos(V ′j − gj) +

∑

k

Ajkajk cos(V ′jk + αjk − gk) +

∑

l

Ajlajk cos(V ′jl + αjl − gjl)

where if

aj < 0, V ′ = V − 3

4, αjk = 1

2, αjl = 3

4,

and if

aj > 0, V ′ = V − 1

4, αjk = 0, αjl = 1

4.

A further phase adjustment to satellite constituents can be made if we wish to ensure thattheir amplitudes are positive. This convention was adopted for the data package of Appendix 7.1(variable PH read in data input (iv) from logical unit 8). Replacing ajk and ajl by their absolutevalues we now see that

αjk = 0 if both ajk and aj have the same sign,

= 1

2otherwise;

αjkl = 1

4if both ajl and aj have the same sign,

= 3

4otherwise.

Similarly, for the slow and semidiurnal constituents, the cluster contribution can be written as

|aj | cos(V ′j − gj) +

∑

k

Ajk|ajk| cos(V ′jk + αjk − gjk) +

∑

l

Ajl|ajl| cos(V ′jl + αjl − gjl),

whereV ′ = V + 1

2if aj < 0,

V otherwise;

αjk = 0 if ajk and aj have the same sign,1

2otherwise;

αjl = − 1

4if ajl and aj have the same sign,

1

4otherwise.

26

Special note should be made of the terdiurnal M3 because both it and its only satellite aredue to third-order terms in the tidal potential. Hence both contribute directly through a cosineterm and so behave as if they were second order semidiurnals.

In order to determine the amplitude and phase of the major contributor, we assume thatthe result actually found in the analysis was fjaj cos(V ′

j − gj + uj), where fj and uj are calledthe nodal modulation corrections in amplitude and phase respectively. To avoid a possiblemisunderstanding, it is worth mentioning here that the term nodal modulation is actually amisnomer. It and the symbols f and u were first used before the advent of modern computersto designate corrections for the moon’s nodal progression that were not incorporated into thecalculations of the astronomical argument for the main constituent. However, now the termsatellite modulation is more appropriate because our correction is due to the presence of satel-lite constituents differing not only in the contribution of the lunar node to their astronomicalargument, but also in the lunar and solar perigee effect.

For the purpose of calculating fj and uj it is assumed that the admittance is very nearlya constant over the frequency range within a constituent cluster, and so gj = gjk = gjl; andrjk = |ajk|/|aj |, rjl = |ajl|/|aj | are equal to the ratio of the tidal equilibrium amplitudes ofthe satellite to the major contributor. These ratios are latitude dependent when satellites ofthe third order are involved, necessitating the correction factors mentioned in Section 2.1.3.However, the ratios are usually small and the correction is slight.

Dropping the ‘prime’ notation and grouping the second- and third-order terms in one sum-mation, the relationship between the analysed results for a main constituent and the actualcluster contribution is

fj |Aj | cos(Vj + uj − gj) = |aj |[

cos(Vj − gj) +∑

k

Ajkrjk cos(Vj − gj + ∆jk + αjk)]

,

where ∆jk = Vjk − Vj .

Expanding this result and observing that it must be true for all Vj(t), the following explicitformulae are found for f and u:

fj =

[

(

1 +∑

k

Ajkrjk cos(∆jk + αjk))2

+(

∑

k

Ajkrjk sin(∆jk + αjk))2

]1/2

,

uj = arctan

[ ∑

k Ajkrjk sin(∆jk + αjk)

1 +∑

k Ajkrjk cos(∆jk + αjk)

]

.

For an analysis carried out over 2N + 1 consecutive observations, ∆t time units apart, Ajk isgiven by

Ajk =sin[(2N + 1)∆t(σjk − σj)/2]

(2N + 1) sin[∆t(σjk − σj)/2],

where σj is the frequency of the main contributor and σjk is that of its satellite. However, Ajk

is very nearly one, even for a one-year analysis, and in the program it is approximated by thisvalue.

For a shallow water constituent whose frequency is calculated as∑N0

j=1cjσj , where σj is the

frequency of the jth main constituent from which it is derived and cj is the linear coefficient,the nodal modulation corrections for amplitude and phase are computed as

f =

N0∏

j=1

f|cj |j and u =

N0∑

j=1

cjuj .

27

2.3.3 Final amplitude and phase results

The result of the least squares analysis was to find for a constituent with frequency σj ,the optimal amplitude Aj and phase φj value for the tidal signal Aj cos 2π(σjt− φj). However,due to nodal corrections, when the astronomical argument is calculated at the central timeorigin t = 0 of the record, we know that the actual contribution of the constituent clusteris fjaj cos 2π(Vj + uj − gj). Hence the amplitude and Greenwich phase lag of the constituentcorresponding to frequency σj can be calculated as aj = Aj/fj and gj = Vj + uj + φj .

2.3.4 Inferred constituents

In accordance with previous notation, tidal signals in this section are assumed to be realin nature. However, an alternative presentation using complex numbers and the basis for thefollowing development is given by Godin (1972).

If the length of a specific tidal record is such that certain important constituents will notbe included directly in the analysis, provision is made via the data input on logical unit 4 toinclude these constituents indirectly by inferring their amplitudes and phases from neighbouringconstituents that are included. If accurate amplitude ratios and phase differences are specified,inference has the effect of significantly reducing any periodic behaviour in the amplitudes andphases of the constituent used for the inference. This is due to the removal of interaction fromthe neighbouring inferred constituent. If it so happens that a constituent specified for inferenceis included directly in the analysis, the program will ignore the inference calculations.

The actual adjustments are as follows. Assume that the constituent with frequency, σ2, isto be inferred from the constituent with frequency, σ1, and that the least squares fit analysisfound the latter’s contribution to be A0

1 cos 2π(σ1t − φ01), where A0

1 and φ01 are the amplitude

and phase respectively (σ1 and φ01 are measured in cycles/h and cycles respectively). Letting

V U1 be the astronomical argument + nodal modulation phase correction,g1 be the Greenwich phase lag,f1 be the nodal modulation amplitude correction factor,

and a1 be the corrected amplitude.

then from Section 2.3.3 we know that

−φ1 = V U1 − g1

and

a1 = A1/f1.

Assuming that A1 and φ1 are the post-inference amplitude and phase respectively for the con-stituent with frequency, σ1,

r12 =a2

a1

=(A2/f2)

(A1/f1)

and

ζ = g1 − g2 = V U1 + φ1 − V U2 − φ2

(the latter two being data input variables R and ZETA respectively), then the presence of the

inferred constituent in the analysed signal yields the relationship:

A01 cos 2π(σ1t − φ0

1) = A1 cos 2π(σ1t − φ1) + A2 cos 2π(σ2t − φ2)

28

= A1 cos 2π(σ1t − φ1){

1 + r12

(

f2

f1

)

cos 2π[(σ2 − σ1)t + V U2 − V U1 + ζ]

}

− A1 sin 2π(σ1t − φ1){

r12

(

f2

f1

)

sin 2π[(σ2 − σ1)t + V U2 − V U1 + ζ]

}

.

Since the constituent with frequency σ2 was not chosen for inclusion in the least squares analysis,|σ2 − σ1|N < RAY , where N is the record length in hours and RAY is the Rayeigh criterionconstant (usually 1.0). Assuming in general that |σ2 − σ1|N is small, good approximations tocos 2π[(σ2−σ1)t+V U2−V U1 + ζ] and sin 2π[(σ2−σ1)t+V U2−V U1 + ζ] are their average valuesover the interval [−N/2, N/2], namely sin[πN(σ2−σ1)] cos[2π(V U2−V U1+ζ)]/[πN(σ2−σ1)] andsin[πN(σ2 − σ1)] sin[2π(V U2 − V U1 + ζ)]/[πN(σ2 − σ1)] respectively. Making these substitutionsand setting

S = r12

(

f2

f1

)

sin[πN(σ2 − σ1)] sin[2π(V U2 − V U1 + ζ)]/[πN(σ2 − σ1)]

and

C = 1 + r12

(

f2

f1

)

sin[πN(σ2 − σ1)] cos[2π(V U2 − V U1 + ζ)]/[πN(σ2 − σ1)],

we obtainA0

1

A1

cos[2π(σ1t − φ01)] = C cos[2π(σ1t − φ1)] − S sin[2π(σ1t − φ1)].

Expanding and regrouping this result yields

cos 2πσ1t

(

A01

A1

cos 2πφ01 − C cos 2πφ1 − S sin 2πφ1

)

= sin 2πσ1t

(

−A01

A1

sin 2πφ01 + C sin 2πφ1 − S cos 2πφ1

)

.

Now since this relationship must hold for all t, both terms in brackets are equal to zero.

HenceA0

1

A1

cos 2πφ01 = C cos 2πφ1 + S sin 2πφ1,

A01

A1

sin 2πφ01 = C sin 2πφ1 − S cos 2πφ1

and so

A1 =A0

1√C2 + S2

,

φ1 = φ01 +

arctan(S/C)

2π.

The relative phase and amplitude of the inferred constituent are then calculated as

φ2 = V U1 − V U2 + φ1 − ζ

and

A2 = r12A1

(

f2

f1

)

.

29

3 USE OF THE TIDAL HEIGHTS PREDICTION

COMPUTER PROGRAM

3.1 General Description

This program produces tidal height values at a given location for a specified period of time.Amplitudes and Greenwich phase lags of the tidal constituents to be used in the predictionare required as input and either equally spaced heights or all the high and low values can beproduced.

3.2 Routines Required

(1) MAIN . . . . . . reads in tidal station and time period information, amplitudes andGreenwich phases of constituents to be used in the prediction, andcalculates the desired tidal heights.

(2) ASTRO . . . . . . reads the standard constituent data package and calculates thefrequencies, astronomical arguments, and nodal corrections for allconstituents.

(3) PUT . . . . . . controls the output for high–low predictions.

(4) HPUT . . . . . . controls the output for equally spaced predictions.

(5) GDAY . . . . . . returns the consecutive day number from a specific origin for anygiven date and vice versa.

(6) ASTR . . . . . . calculates ephermides for the sun and moon.

3.3 Data Input

All input data required by the tidal heights prediction program is from logical unit 8. A sampleset is given in Appendix 7.4. Although data types (i), (ii) and (iii) are identical to types (ii),(iii) and (iv) expected in logical unit 8 by the analysis program, for completeness they arerepeated here.

(i) Two cards specifying values for the astronomical arguments SO,HO,PO,ENPO,PPO,DS,DH,DP,

DNP,DPP in the format (5F13.10).

SO = mean longitude of the moon (cycles) at the reference time origin;HO = mean longitude of the sun (cycles) at the reference time origin;PO = mean longitude of the lunar perigee (cycles) at the reference time origin;

ENPO = negative of the mean longitude of the ascending node (cycles) at the referencetime origin;

PPO = mean longitude of the solar perigee (perihelion) at the reference time origin.

30

DS,DH,DP,DNP,DP are their respective rates of change over a 365-day period at the referencetime origin.

Although these argument values are not used by the program that was revised inOctober 1992, in order to maintain consistency with earlier programs, they are still requiredas input. Polynomial approximations are now employed to more accurately evaluate theastronomical arguments and their rates of change.

(ii) At least one card for all the main tidal constituents specifying their Doodson numbers andphase shift, along with as many cards as are necessary for the satellite constituents. Thefirst card for each such constituent is in the format (6X,A5,1X,6I3,F5.2,I4) and containsthe following information:

KON = constituent name;II,JJ,KK,LL,MM,NN = the six Doodson numbers for KON;

SEMI = phase correction for KON;NJ = number of satellite constituents.

A blank card terminates this data type.

If NJ>0, information on the satellite constituents follows, three satellites per card, inthe format (11X,3(3I3,F4.2,F7.4,IX,I1,1X)). For each satellite the values read are:

LDEL,MDEL,NDEL = the last three Doodson numbers of the main constituent subtractedfrom the last three Doodson numbers of the satellite constituent;

PH = phase correction of the satellite constituent relative to the phase ofthe main constituent;

EE = amplitude ratio of the satellite tidal potential to that of the mainconstituent;

IR = 1 if the amplitude ratio has to be multiplied by the latitudecorrection factor for diurnal constituents,

= 2 if the amplitude ratio has to be multiplied by the latitudecorrection factor for semidiurnal constituents,

= otherwise if no correction is required to the amplitude ratio.

(iii) One card specifying each of the shallow water constituents and the main constituents fromwhich they are derived. The format is (6X,A5,I1,2X,4(F5.2,A5,5X)) and the respectivevalues read are:

KON = name of the shallow water constituent;NJ = number of main constituents from which it is derived;

COEF,KONCO = combination number and name of these main constituents.

The end of these shallow water constituents is denoted by a blank card.

(iv) One card with the tidal station information ISTN,(NA(J),J=1,4),ITZONE,LAD,LAM,LOD,

LOM in the format (5X,I4,1X,3A6,A4,A3,1X,I2,1X,I2,2X,I3,1X,I2).

ISTN = station number;(NA(J),J=1,4) = station name;

ITZONE = time zone reference for the “Greenwich” phases;

31

LAD,LAM = station latitude in degrees and minutes;LOD,LOM = station longitude in degrees and minutes.

(v) One card for each constituent to be included in the prediction with the constituent name(KON), amplitude (AMP) and phase lag (G) in the format (5X,A5,28X,F8.4,F7.2). (Thisformat is compatible with the analysis program results produced on output device 2). Thephase lag units should be degrees (measured in time zone ITZONE while the units of the pre-dicted tidal heights will be the same as those of the input amplitudes. The last constituentis followed by a blank card.

(vi) One card containing the following information on the period and type of prediction desired.The format is (3I3,1X,3I3,1X,A4,F9.5,2X,2I3).

IDYO,IMOO,IYRO = first day, month and year of the prediction period;IDYE,IMOE,IYRE = first day, month and year of the prediction period;

ITYPE = ′EQUI′ if equally spaced predictions are desired,= ′EXTR′ if all the high and low tide times and heights are desired;

DT = time spacing of the predicted values if ITYPE=′EQUI′,= time step increment used to initially bracket a high or low value if

ITYPE=′EXTR′;ICE0,ICEE = century number for the beginning and end of the prediction.

(Blank values for ICE0 or ICEE will be reset to 19.)

Equally spaced predictions begin at DT hours on the first day and extend to 2400 h (assum-ing 24 is a multiple of DT) of the last day. When ITYPE=′EXTR′, Godin and Taylor (1973)recommend using the following values for DT: 3 h for a semidiurnal tide, 6 h for a diurnaltide and 0.5 h for a mixed tide.

Type (vi) data may be repeated any number of times. One blank card following a type(vi) record will return the program to type (iv) input, while two blank cards will end theprogram execution.

3.4 Output

Two logical units are used for the output of results in the tidal heights prediction program.Device number 6 is the line printer and 10 is a data file. Both equally spaced and high–low predictions are put onto both devices with the same format. However the line printeralso records the station name and location along with the amplitudes and phase lags of theconstituents used in the prediction. Appendix 7.5 lists device 10 output resulting from the inputof Appendix 7.4.

When daily high–low values are desired, the date, station number and a series of up to sixheights and occurrence times are listed per record. Each record begins with the variable HLwhose value is zero if the first height for that day is a high (i.e. larger than the second height)and one if the first height is a low. If there are less than six high–low values for a day, theyare padded up to six with the values 9999 and 99.9 for the times and heights respectively. Ondevice 10, the format used for the variables HL, the station number, the day, month, year, andthe six pairs of times and heights is (1X,I1,I5,2I3,I2,6(I5,F5.1)).

When equally spaced heights are requested, 8 values are listed on each record precededby the station number, the time, day, month and year of the first value, and followed by

32

the time increment between heights. On device number 10, the format for these variables is(1X,I4,F8.4,I3,2I2,8F6.3,F12.4)

3.5 Program Conversion, Modifications, Storage and Dimension Guidelines

The source program and constituent data package described in this manual have beentested on various mainframe, PC and workstation computers at the Institute of Ocean Sciences,Patricia Bay. Although as much of the program as possible was written in basic FORTRAN,some changes may be required before the program and data package can be used on otherinstallations. Please write or call the author if any problems are encountered.

The program in its present form requires approximately 33,000 bytes for the storage of itsinstructions and arrays respectively. As with the analysis program, changing the number ortype of constituents in the standard data package may require alteration to the dimensions ofsome arrays. Restrictions on the minimum dimension of all arrays are now given.

Let

MTAB be the total number of possible constituents contained in the data package(presently 146),

M be the number of constituents to be included in the prediction,

MCON be the number of main constituents in the standard data package (presently 45),

MSAT be the sum of the number of satellites for these main constituents and the numberof main constituents with no satellites (presently 162 plus 8 for the version of theconstituent data package, listed in Appendix 7.4, that contains no third-ordersatellites for both N2 and L2),

MSHAL be the sum for all shallow water constituents of the number of main constituentsfrom which each is derived (presently 251),

NITER be the iterations required to reduce the time interval within which it is knownthat a high or low tide exists, to a desired length (with the largest initialinterval size of 6 h and a 6-min final interval, NITER is 6).

Then in the main program, arrays SIGTAB,V,U and F should have minimum dimension MTAB;array KONTAB should have minimum dimension MTAB+1; arrays SIG,INDX,TWOC,CH,CHP,CHA,

CHB,CHM,ANGO and AMPNC should have minimum dimension M; arrays KON,AMP and G shouldhave minimum dimension M+1; and the two-dimensional array BTWDC should have a minimumdimension of M by NITER. Array COSINE which stores pre-calculated cosine function values overthe range of 0◦ to 360◦ and is used as a look-up table, presently has 2002 elements.

In subroutine ASTRO, the arrays FREQ,V,U and F should have minimum dimension MTAB;arrays KON and NJ should have minimum dimension MTAB+1; arrays II,JJ,KK,LL,MM,NN andSEMI should have minimum dimension MCON+1; arrays EE,LDEL,MDEL,NDEL,IR and PH shouldhave minimum dimension MSAT; and arrays KONCO and COEF should have minimum dimensionMSHAL+4.

In subroutine PUT, the dimensions of arrays HGTK and ITIME should be at least as largeas the maximum number of high and low values per day (this is presently assumed to be 9).

In subroutine HPUT, the dimension of array H should be at least equal to the number ofequally spaced tidal height values per output record of logical unit 10 or 6 (presently, this is 8).

In subroutine CDAY, both arrays NDM and NDP should have dimension 12.

33

4 TIDAL HEIGHTS PREDICTION PROGRAM DETAILS

4.1 Problem Formulation and the Equally Spaced Predictions Method

The tidal height, h(t), at a particular station may be represented by the harmonic summa-tion (see Section 2.3.3)

h(t) =m

∑

j=1

fj(t)Aj cos [2π(Vj(t) + uj(t) − gj)] , (1)

whereAj , gj = amplitude and phase lag of constituent, j,

fj(t), uj(t) = nodal modulation amplitude and phase correction factors for constituent, j,Vj(t) = astronomical argument for constituent, j.

Expanding V (t) as in Section 2.3.1 and using the first-order Taylor approximations for theastronomical arguments as in Section 2.1.1, V (t) can be re-expressed as

V (t) = iτ(t) + jS(t) + kH(t) + lP (t) + mN ′(t) + nP ′(t)

= iτ(t0) + jS(t0) + kH(t0) + lP (t0) + mN ′(t0) + nP ′(t0)

+ (t − t0)∂

∂t

[

iτ(t) + jS(t) + kH(t) + lP (t) + mN ′(t) + nP ′(t)]

t=t0

= V (t0) + (t − t0)σ,

where t0 is the reference time origin and σ is the constituent frequency at this time origin. Itfollows from this result that V (t2) = V (t1) + (t2 − t1)σ for arbitrary times, t1, t2, and so Vj(t)can be replaced in (1) by Vj(t1) + (t − t1)σj for some convenient time, t1.

From Section 2.3.2 it is seen that f(t) and u(t) are time dependent only through the ∆jk(t)variable. Since satellites differ from main constituents in only the last three Doodson numbers(see Section 2.1.3),

∆jk(t) = Vjk(t) − Vj(t)

= ∆lP (t) + ∆mN ′(t) + ∆nP ′(t).

Using the first order Taylor approximations for P , N ′ and P ′, it follows that over a time period[t1, t2] the change in ∆jk(t) is

∆jk(t2) − ∆jk(t1) = ∆l[P (t2) − P (t1)] + ∆m[N ′(t2) − N ′(t1)] + ∆n[P ′(t2) − P ′(t1)]

= (t2 − t1)d

dt[∆lP (t) + ∆mN ′(t) + ∆nP ′(t)]t=t0

= (t2 − t1)(σjk − σj).

Since d/dt[P (t) + N ′(t) + P ′(t)]t=t0 is 0.16668884 cycles/356 days and |∆l|, |∆m|, |∆n| arealways less than or equal to 4, if |t2 − t1| ≤ 16 days, |∆jk(t2) − ∆jk(t1)| ≤ 0.03 cycles. Thissmall variation in ∆jk(t) leads to a similar behaviour in cos[∆jk(t)] and sin[∆jk(t)], and hencef(t) and u(t). Thus only a small loss in accuracy but a considerable calculation time saving will

34

result if f(t) and u(t) are approximated by a constant value throughout the period of a month.Consequently f(t) and u(t) are assumed to equal their value at 0000 h of the sixteenth dayof the month for the entire monthly period; for convenience, V (t) is set to V (t16) + (t − t16)σ,where t16 is this same time.

The procedure for calculating a series of tidal heights is then as follows. Since the tidalprediction data package does not contain constituent frequencies, they must be calculated viathe astronomical variable derivatives and the constituent Doodson numbers. The values f , uand V are then calculated for the sixteenth day of the first month of the desired predictionperiod and, as required, for subsequent months. Tidal heights for the desired values of t canthen be calculated as

h(t) =

m∑

j=1

fj(t16)Aj cos[2π(Vj(t16) + (t − t16)σj + uj(t16) − gj)]. (2)

In order to avoid calling a trigonometric library function for each new value of t, when asequence of equally spaced heights are required, the following Chebyshev iteration formula isused for each constituent contribution,

f(n + 1) = 2 cos(σ∆t)f(n) − f(n − 1), (3)

where f(n) = cos(nσ∆t) or sin(nσ∆t).

4.2 The High and Low Tide Prediction Method

The material presented here is taken from Godin and Taylor (1973).

In Section 4.1 we saw that the tidal height at a given location can be represented by theharmonic sum

h(t) =

m∑

j=1

fj(t0)Aj cos[2π(Vj(t0) + (t − t0)σj + u(t0) − gj)] (1)

whereAj , gj , σj = amplitude, phase lag and frequency of constituent, j,

fj(t0), uj(t0) = nodal modulation amplitude and phase correction factors for constituent,j, at the time origin t0,

Vj(t0) = astronomical argument for constituent j at the time origin t0.

Letting D(t) be the derivative of h(t), i.e.

D(t) = −m

∑

j=1

fj(t0)Aj2πσj sin[2π(Vj(t0) + (t − t0)σj + u(t0) − gj)], (2)

the high–low tide prediction method uses the following calculus results. If D(t) is a continuousfunction on the interval [t1, t2] and tk is a point in this interval, then:

(i) D(tk) = 0 if and only if tk is an extreme point or saddle point,5 or h(t) is constant in theneighbourhood of tk;

(ii) if D(t1) and D(t2) have opposite signs, then there exists a tk in (t1, t2) with D(tk) = 0.

5 An example of a saddle point is x = 0 for the function f(x) = x3.

35

Now for computational purposes we can assume that saddle points do not exist. That isto say, due to accuracy limitations of the computer, a zero derivative will be approximated bya number with a very small absolute value and thus perturb a saddle point so that it becomeseither a maximum or minimum, or a near saddle point (in the neighbourhood of a “near saddlepoint”, the derivative is of constant sign and almost assumes the value zero). And since, from itsdefinition, we can reasonably assume that h(t) is not constant over any arbitrarily small interval,the continuity of D(t) everywhere implies that an interval [t1, t2] with D(t1) and D(t2), havingopposite signs, contains an extremum.

However, this result alone is not sufficient to guarantee the location of all extrema becauseit does not eliminate the possibility of having more than one extremum in an interval whoseendpoints have different signs, nor does it imply that if the endpoints have the same derivativesign there is no extremum in the interval. In order to ensure these conditions and thus beassured of bracketing all extreme values, it is necessary that a minimum interval size be specifiedin which we can assume that there exists, at most, one high or low tide.

Clearly, the interval size, ∆t, will be dependent upon the nature of the tide at a particularstation. The time between successive high and low waters for predominantly semidiurnal anddiurnal tides is approximately 6 and 12 h respectively. However, if the tide is mixed, the patternof extremes is more complicated. Figure 2 shows the water level at Victoria, British Columbiabetween July 24 and 31, 1976. It is a mixed tide where the shorter period fluctuations overridethe major diurnal oscillations with a continuous shift in their position and amplitude.

One characterization of the tide may be obtained by calculating the ratio of the amplitudesof the major harmonic constituents, M2, S2 K1 and O1. This value is called the form number(Dietrich, 1963) and is defined precisely as

F =K1 + O1

M2 + S2

.

The tide is then said to be

(i) semidiurnal if 0 ≤ F ≤ 0.25,

(ii) mixed if 0.25 < F ≤ 3.00,

(iii) diurnal if F > 3.00.

For Victoria, F = 2.1.

In accordance with this determination, Godin suggests the following maximum time intervalvalues in which it can be assumed that there exists at most one extremum:

(i) ∆t = 3 h for semidiurnal tide,

(ii) ∆t = 0.5 h for mixed tide,

(iii) ∆t = 6 h for diurnal tide.

Although in fact, a mixed tide may have extrema closer than 0.5 h, he feels that forpractical purposes it is sufficient to note just one of them.

With these values of ∆t we can then bracket all extrema by moving forward in time withsteps of size, ∆t, and comparing signs of the interval endpoints. Once such upper and lowerbounds have been found, the extreme point can be located exactly by any one of a number ofsearch techniques. Because it requires a minimal amount of time, the one chosen is Bolzano’smethod of bisection coupled with linear interpolation. Although the bisection method does nottake the minimal number of iterations when compared to more sophisticated search techniques,

36

Figure 2 Synthesized water level at Victoria, British Columbia over the period July 24 to31, 1976. The tide is of a mixed character with F = 2.1. The arrows indicate the time andheight of the extrema predicted using the method described in Section 4.2. (Redrawn from C.Wallace)

it is able to make significant time savings by computing new sine function values as a linearcombination of old ones and thus, unlike the other methods, avoid calls to the FORTRANlibrary function SIN.

10

8

6

4

2

024 25 26 27 28 29 30 31

July 1976

Tid

al heig

ht (f

eet)

In more detail, the search algorithm for an extremum is then as follows:

(i) Move forward in time from the origin, or the last extremum, in steps of ∆t until either achange in sign exists between the derivative values at the endpoints of the interval (ta, tb),or tb extends beyond the desired prediction period. Each constituent contribution in thesummation D(t) is evaluated by the Chebyshev iteration formula (3) of Section 4.1. Whenan interval containing an extremum is located, set k = 1 and proceed to (ii).

(ii) Calculate tk = ta + 1

2k ∆t and for each constituent in the sum evaluate D(tk) by using theformula

sin(tk) =sin(ta) + sin(tb)

2 cos(1/2k∆t).

If |D(tk)| ≤ 10−16, set D(tk) = 10−16.

37

Figure 3 An example of the sequence of steps involved in locating a zero tk of the derivative,D(t). The sign of D(t) at the various points tested is denoted by a plus or minus. After astep, ∆t, the sign has changed; by a retrogression of 1

2∆t, the sign has reverted to plus, forcing

a forward step of 1

4∆t where the sign is still unchanged. Two further forward steps of 1

8∆t and

1

16∆t locate the minimum width interval (tr, tr+1) over which the position of tk is determined

by linear interpolation from the values of D(t) at tr and tr+1. (Redrawn from C. Wallace)

(iii) Re-assign whichever of ta or tb has the same derivative sign as D(tk), by tk. If the newinterval length tb− ta is less than 0.1 h, proceed to (iv). Otherwise set k = k+1 and returnto (ii).

(iv) Use the following linear interpolation formula to find the extremum tE ,

tE = ta + [D(ta)(tb − ta)]/[D(ta) − D(tb)],

and evaluate h(tE) via (1). For each constituent term in this sum, obtain the functionvalue by using a pre-calculated stored table of 2002 cosine values with arguments in therange of 0◦ to 360◦. Return to (i).

Figure 3 illustrates an example of the sequence of steps involved in the search for an extremevalue. It is easily calculated that the number of iterations required to reduce the bracketinginterval from ∆t to 0.1 h is six for diurnal tides, three for mixed tides, and five for semidiurnaltides.

∆t

1/2 ∆t

1/4 ∆t

1/8 ∆t

1/16 ∆t

• •• • • •++++

t r tk

– –

t r +1

tkD ( ) = 0t

Arrows in Figure 2 indicate the extrema predicted for Victoria using the technique justdescribed; the shaft of the arrow locates the time abscissa while the tip ends at the predictedheight. The predicted hourly heights and the times and heights of all extrema are listed inAppendix 7.5.

38

5 CONSISTENCY OF THE ANALYSIS AND PREDICTION

PROGRAMS

Although consistency between the tidal heights analysis program and the tidal heightsprediction program was a major objective in their revision, they do have one difference. Inparticular, if a pseudo-tidal record were generated by the prediction program and analysedusing the same constituents, the amplitude and phase results given by the analysis programwould not be identical to those used as input for the prediction program.

In a small part, this discrepancy is due to round-off accumulated during the calculations.However, a test performed at the Institute of Ocean Sciences with a two-month period ofsynthesized hourly heights indicates that such errors occur no sooner than the fourth digit. Theremainder of the difference (which is, at worst, in the third digit) can be attributed to differentapproximating assumptions for the calculation of f and u, the nodal modulation amplitudeand phase correction factors. Whereas the prediction program calculates these values at thesixteenth day of each month in the desired time period and keeps them constant throughoutthe entire month, the analysis program assumes them to be constant over the entire analysisperiod and equal to their true values at the central hour of that period.

It is important to note, though, that significantly different results can be expected in asimilar test run if there is at least one more constituent used in the synthesis than analysis. Thisis because the least squares fit technique will adjust the amplitudes and phases of constituentsincluded in the analysis to partially account for contributions due to constituents included inthe synthesis but not the analysis. In fact, this will occur even if the extra constituents areinferred (e.g. P1 is included in the synthesis and in the analysis via inference from K1) becauseof small inaccuracies in the approximating inference assumptions. However, except for round-offerrors and the slightly different f and u values, having more constituents in the analysis thanthe synthesis will not affect the results.

39

6 REFERENCES

Cartwright, D.E. and R.J. Tayler, 1971. New computations of the tide-generating potential.Geophys. J. Roy. Astron. Soc. 23: 45–74.

Cartwright, D.E. and A.C. Edden, 1973. Corrected tables of tidal harmonics. Geophys. J. Roy.

Astron. Soc. 33: 253–264.

Davis, P.J. and P. Rabinowitz, 1961. Advances in orthonormalizing computation. Advances in

Computing. 2: 56–57.

Dietrich, G., 1963. General Oceanography. Interscience Publishers. New York.

Doodson, A.T., 1921. The harmonic development of the tide-generating potential. Proc. Roy.

Soc. Series A. 100: 306–323. Re-issued in the International Hydrographic Review, May1954.

Faddeev, D.K. and V.N. Faddeeva, 1963. Computational Methods of Linear Algebra. W.H.Freeman and Company, San Francisco.

Foreman, M.G.G., 1978. Manual for Tidal Currents Analysis and Prediction. Pacific MarineScience Report 78-6, Institute of Ocean Sciences, Patricia Bay, Victoria, B.C. x pp.

Forsythe, G.E. and C.B. Moler, 1967. Computer Solution of Linear Algebraic Systems. Prentice-Hall, Englewood Cliffs, N.J.

Godin, G., 1972. The Analysis of Tides. University of Toronto Press, Toronto.

Godin, G. and J. Taylor, 1973. A simple method for the prediction of the time and height ofhigh and low water. Reprint from the International Hydrographical Review, Vol. L, No.2,July 1973.

Godin, G., 1974. Nodal corrections. (unpublished notes).

Her Majesty’s Nautical Almanac Office, 1961. Explanatory Supplement to the AstronomicalEphemeris and the American Ephemeris and Nautical Almanac. Her Majesty’s StationeryOffice, London.

Meeus, J. 1988. Astronomical Formulae for Calculators. Willmann-Bell, Richmond, VA.

Searle, S.R., 1971. Linear Models. John Wiley and Sons, New York.

Wilkinson, J.H., 1967. The Solution of Ill-Conditioned Linear Equations in Mathematical Meth-ods for Digital Computers, Vol. 2, edited by Ralston and Wilf.

40

Appendix 7.1 Standard Constituent Input Data for the Tidal Heights Analysis Computer Program.This Data is Read by the Program from Logical Unit 8.

Z0 0.0 M2

SA 0.0001140741 SSA

SSA 0.0002281591 Z0

MSM 0.0013097808 MM

MM 0.0015121518 MSF

MSF 0.0028219327 Z0

MF 0.0030500918 MSF

ALP1 0.0343965699 2Q1

2Q1 0.0357063507 Q1

SIG1 0.0359087218 2Q1

Q1 0.0372185026 O1

RHO1 0.0374208736 Q1

O1 0.0387306544 K1

TAU1 0.0389588136 O1

BET1 0.0400404353 NO1

NO1 0.0402685944 K1

CHI1 0.0404709654 NO1

PI1 0.0414385130 P1

P1 0.0415525871 K1

S1 0.0416666721 K1

K1 0.0417807462 Z0

PSI1 0.0418948203 K1

PHI1 0.0420089053 K1

THE1 0.0430905270 J1

J1 0.0432928981 K1

2PO1 0.0443745198

SO1 0.0446026789 OO1

OO1 0.0448308380 J1

UPS1 0.0463429898 OO1

ST36 0.0733553835

2NS2 0.0746651643

ST37 0.0748675353

ST1 0.0748933234

OQ2 0.0759749451 EPS2

EPS2 0.0761773161 2N2

ST2 0.0764054753

ST3 0.0772331498

O2 0.0774613089

2N2 0.0774870970 MU2

MU2 0.0776894680 N2

SNK2 0.0787710897

N2 0.0789992488 M2

NU2 0.0792016198 N2

ST4 0.0794555670

OP2 0.0802832416

GAM2 0.0803090296 H1

H1 0.0803973266 M2

M2 0.0805114007 Z0

H2 0.0806254748 M2

MKS2 0.0807395598 M2

ST5 0.0809677189

ST6 0.0815930224

LDA2 0.0818211815 L2

L2 0.0820235525 S2

41

2SK2 0.0831051742

T2 0.0832192592 S2

S2 0.0833333333 M2

R2 0.0834474074 S2

K2 0.0835614924 S2

MSN2 0.0848454852 ETA2

ETA2 0.0850736443 K2

ST7 0.0853018034

2SM2 0.0861552660

ST38 0.0863576370

SKM2 0.0863834251

2SN2 0.0876674179

NO3 0.1177299033

MO3 0.1192420551 M3

M3 0.1207671010 M2

NK3 0.1207799950

SO3 0.1220639878 MK3

MK3 0.1222921469 M3

SP3 0.1248859204

SK3 0.1251140796 MK3

ST8 0.1566887168

N4 0.1579984976

3MS4 0.1582008687

ST39 0.1592824904

MN4 0.1595106495 M4

ST9 0.1597388086

ST40 0.1607946422

M4 0.1610228013 M3

ST10 0.1612509604

SN4 0.1623325821 M4

KN4 0.1625607413

MS4 0.1638447340 M4

MK4 0.1640728931 MS4

SL4 0.1653568858

S4 0.1666666667 MS4

SK4 0.1668948258 S4

MNO5 0.1982413039

2MO5 0.1997534558

3MP5 0.1999816149

MNK5 0.2012913957

2MP5 0.2025753884

2MK5 0.2028035475 M4

MSK5 0.2056254802

3KM5 0.2058536393

2SK5 0.2084474129 2MK5

ST11 0.2372259056

2NM6 0.2385098983

ST12 0.2387380574

2MN6 0.2400220501 M6

ST13 0.2402502093

ST41 0.2413060429

M6 0.2415342020 2MK5

MSN6 0.2428439828

MKN6 0.2430721419

ST42 0.2441279756

2MS6 0.2443561347 M6

42

2MK6 0.2445842938 2MS6

NSK6 0.2458940746

2SM6 0.2471780673 2MS6

MSK6 0.2474062264 2SM6

S6 0.2500000000

ST14 0.2787527046

ST15 0.2802906445

M7 0.2817899023

ST16 0.2830867891

3MK7 0.2833149482 M6

ST17 0.2861368809

ST18 0.3190212990

3MN8 0.3205334508

ST19 0.3207616099

M8 0.3220456027 3MK7

ST20 0.3233553835

ST21 0.3235835426

3MS8 0.3248675353

3MK8 0.3250956944

ST22 0.3264054753

ST23 0.3276894680

ST24 0.3279176271

ST25 0.3608020452

ST26 0.3623141970

4MK9 0.3638263489

ST27 0.3666482815

ST28 0.4010448515

M10 0.4025570033

ST29 0.4038667841

ST30 0.4053789360

ST31 0.4069168759

ST32 0.4082008687

ST33 0.4471596822

M12 0.4830684040

ST34 0.4858903367

ST35 0.4874282766

.7428797055 .7771900329 .5187051308 .3631582592 .7847990160 000GMT 1/1/76

13.3594019864 .9993368945 .1129517942 .0536893056 .0000477414 INCR./365DAYS

Z0 0 0 0 0 0 0 0.0 0

SA 0 0 1 0 0 -1 0.0 0

SSA 0 0 2 0 0 0 0.0 0

MSM 0 1 -2 1 0 0 .00 0

MM 0 1 0 -1 0 0 0.0 0

MSF 0 2 -2 0 0 0 0.0 0

MF 0 2 0 0 0 0 0.0 0

ALP1 1 -4 2 1 0 0 -.25 2

ALP1 -1 0 0 .75 0.0360R1 0 -1 0 .00 0.1906

2Q1 1 -3 0 2 0 0-0.25 5

2Q1 -2 -2 0 .50 0.0063 -1 -1 0 .75 0.0241R1 -1 0 0 .75 0.0607R1

2Q1 0 -2 0 .50 0.0063 0 -1 0 .0 0.1885

SIG1 1 -3 2 0 0 0-0.25 4

SIG1 -1 0 0 .75 0.0095R1 0 -2 0 .50 0.0061 0 -1 0 .0 0.1884

SIG1 2 0 0 .50 0.0087

Q1 1 -2 0 1 0 0-0.25 10

Q1 -2 -3 0 .50 0.0007 -2 -2 0 .50 0.0039 -1 -2 0 .75 0.0010R1

43

Q1 -1 -1 0 .75 0.0115R1 -1 0 0 .75 0.0292R1 0 -2 0 .50 0.0057

Q1 -1 0 1 .0 0.0008 0 -1 0 .0 0.1884 1 0 0 .75 0.0018R1

Q1 2 0 0 .50 0.0028

RHO1 1 -2 2 -1 0 0-0.25 5

RHO1 0 -2 0 .50 0.0058 0 -1 0 .0 0.1882 1 0 0 .75 0.0131R1

RHO1 2 0 0 .50 0.0576 2 1 0 .0 0.0175

O1 1 -1 0 0 0 0-0.25 8

O1 -1 0 0 .25 0.0003R1 0 -2 0 .50 0.0058 0 -1 0 .0 0.1885

O1 1 -1 0 .25 0.0004R1 1 0 0 .75 0.0029R1 1 1 0 .25 0.0004R1

O1 2 0 0 .50 0.0064 2 1 0 .50 0.0010

TAU1 1 -1 2 0 0 0-0.75 5

TAU1 -2 0 0 .0 0.0446 -1 0 0 .25 0.0426R1 0 -1 0 .50 0.0284

TAU1 0 1 0 .50 0.2170 0 2 0 .50 0.0142

BET1 1 0 -2 1 0 0 -.75 1

BET1 0 -1 0 .00 0.2266

NO1 1 0 0 1 0 0-0.75 9

NO1 -2 -2 0 .50 0.0057 -2 -1 0 .0 0.0665 -2 0 0 .0 0.3596

NO1 -1 -1 0 .75 0.0331R1 -1 0 0 .25 0.2227R1 -1 1 0 .75 0.0290R1

NO1 0 -1 0 .50 0.0290 0 1 0 .0 0.2004 0 2 0 .50 0.0054

CHI1 1 0 2 -1 0 0-0.75 2

CHI1 0 -1 0 .50 0.0282 0 1 0 .0 0.2187

PI1 1 1 -3 0 0 1-0.25 1

PI1 0 -1 0 .50 0.0078

P1 1 1 -2 0 0 0-0.25 6

P1 0 -2 0 .0 0.0008 0 -1 0 .50 0.0112 0 0 2 .50 0.0004

P1 1 0 0 .75 0.0004R1 2 0 0 .50 0.0015 2 1 0 .50 0.0003

S1 1 1 -1 0 0 1-0.75 2

S1 0 0 -2 .0 0.3534 0 1 0 .50 0.0264

K1 1 1 0 0 0 0-0.75 10

K1 -2 -1 0 .0 0.0002 -1 -1 0 .75 0.0001R1 -1 0 0 .25 0.0007R1

K1 -1 1 0 .75 0.0001R1 0 -2 0 .0 0.0001 0 -1 0 .50 0.0198

K1 0 1 0 .0 0.1356 0 2 0 .50 0.0029 1 0 0 .25 0.0002R1

K1 1 1 0 .25 0.0001R1

PSI1 1 1 1 0 0 -1-0.75 1

PSI1 0 1 0 .0 0.0190

PHI1 1 1 2 0 0 0-0.75 5

PHI1 -2 0 0 .0 0.0344 -2 1 0 .0 0.0106 0 0 -2 .0 0.0132

PHI1 0 1 0 .50 0.0384 0 2 0 .50 0.0185

THE1 1 2 -2 1 0 0 -.75 4

THE1 -2 -1 0 .00 .0300 -1 0 0 .25 0.0141R1 0 -1 0 .50 .0317

THE1 0 1 0 .00 .1993

J1 1 2 0 -1 0 0-0.75 10

J1 0 -1 0 .50 0.0294 0 1 0 .0 0.1980 0 2 0 .50 0.0047

J1 1 -1 0 .75 0.0027R1 1 0 0 .25 0.0816R1 1 1 0 .25 0.0331R1

J1 1 2 0 .25 0.0027R1 2 0 0 .50 0.0152 2 1 0 .50 0.0098

J1 2 2 0 .50 0.0057

OO1 1 3 0 0 0 0-0.75 8

OO1 -2 -1 0 .50 0.0037 -2 0 0 .0 0.1496 -2 1 0 .0 0.0296

OO1 -1 0 0 .25 0.0240R1 -1 1 0 .25 0.0099R1 0 1 0 .0 0.6398

OO1 0 2 0 .0 0.1342 0 3 0 .0 0.0086

UPS1 1 4 0 -1 0 0 -.75 5

UPS1 -2 0 0 .00 0.0611 0 1 0 .00 0.6399 0 2 0 .00 0.1318

UPS1 1 0 0 .25 0.0289R1 1 1 0 .25 0.0257R1

OQ2 2 -3 0 3 0 0 0.0 2

OQ2 -1 0 0 .25 0.1042R2 0 -1 0 .50 0.0386

EPS2 2 -3 2 1 0 0 0.0 3

EPS2 -1 -1 0 .25 0.0075R2 -1 0 0 .25 0.0402R2 0 -1 0 .50 0.0373

44

2N2 2 -2 0 2 0 0 0.0 4

2N2 -2 -2 0 .50 0.0061 -1 -1 0 .25 0.0117R2 -1 0 0 .25 0.0678R2

2N2 0 -1 0 .50 0.0374

MU2 2 -2 2 0 0 0 0.0 3

MU2 -1 -1 0 .25 0.0018R2 -1 0 0 .25 0.0104R2 0 -1 0 .50 0.0375

N2 2 -1 0 1 0 0 0.0 4

N2 -2 -2 0 .50 0.0039 -1 0 1 .00 0.0008 0 -2 0 .00 0.0005

N2 0 -1 0 .50 0.0373

NU2 2 -1 2 -1 0 0 0.0 4

NU2 0 -1 0 .50 0.0373 1 0 0 .75 0.0042R2 2 0 0 .0 0.0042

NU2 2 1 0 .50 0.0036

GAM2 2 0 -2 2 0 0 -.50 3

GAM2 -2 -2 0 .00 0.1429 -1 0 0 .25 0.0293R2 0 -1 0 .50 0.0330

H1 2 0 -1 0 0 1-0.50 2

H1 0 -1 0 .50 0.0224 1 0 -1 .50 0.0447

M2 2 0 0 0 0 0 0.0 9

M2 -1 -1 0 .75 0.0001R2 -1 0 0 .75 0.0004R2 0 -2 0 .0 0.0005

M2 0 -1 0 .50 0.0373 1 -1 0 .25 0.0001R2 1 0 0 .75 0.0009R2

M2 1 1 0 .75 0.0002R2 2 0 0 .0 0.0006 2 1 0 .0 0.0002

H2 2 0 1 0 0 -1 0.0 1

H2 0 -1 0 .50 0.0217

LDA2 2 1 -2 1 0 0-0.50 1

LDA2 0 -1 0 .50 0.0448

L2 2 1 0 -1 0 0-0.50 5

L2 0 -1 0 .50 0.0366 2 -1 0 .00 0.0047 2 0 0 .50 0.2505

L2 2 1 0 .50 0.1102 2 2 0 .50 0.0156

T2 2 2 -3 0 0 1 0.0 0

S2 2 2 -2 0 0 0 0.0 3

S2 0 -1 0 .0 0.0022 1 0 0 .75 0.0001R2 2 0 0 .0 0.0001

R2 2 2 -1 0 0 -1-0.50 2

R2 0 0 2 .50 0.2535 0 1 2 .0 0.0141

K2 2 2 0 0 0 0 0.0 5

K2 -1 0 0 .75 0.0024R2 -1 1 0 .75 0.0004R2 0 -1 0 .50 0.0128

K2 0 1 0 .0 0.2980 0 2 0 .0 0.0324

ETA2 2 3 0 -1 0 0 0.0 7

ETA2 0 -1 0 .50 0.0187 0 1 0 .0 0.4355 0 2 0 .0 0.0467

ETA2 1 0 0 .75 0.0747R2 1 1 0 .75 0.0482R2 1 2 0 .75 0.0093R2

ETA2 2 0 0 .50 0.0078

M3 3 0 0 0 0 0 -.50 1

M3 0 -1 0 .50 .0564

2PO1 2 2.0 P1 -1.0 O1

SO1 2 1.0 S2 -1.0 O1

ST36 3 2.0 M2 1.0 N2 -2.0 S2

2NS2 2 2.0 N2 -1.0 S2

ST37 2 3.0 M2 -2.0 S2

ST1 3 2.0 N2 1.0 K2 -2.0 S2

ST2 4 1.0 M2 1.0 N2 1.0 K2 -2.0 S2

ST3 3 2.0 M2 1.0 S2 -2.0 K2

O2 1 2.0 O1

ST4 3 2.0 K2 1.0 N2 -2.0 S2

SNK2 3 1.0 S2 1.0 N2 -1.0 K2

OP2 2 1.0 O1 1.0 P1

MKS2 3 1.0 M2 1.0 K2 -1.0 S2

ST5 3 1.0 M2 2.0 K2 -2.0 S2

ST6 4 2.0 S2 1.0 N2 -1.0 M2 -1.0 K2

2SK2 2 2.0 S2 -1.0 K2

45

MSN2 3 1.0 M2 1.0 S2 -1.0 N2

ST7 4 2.0 K2 1.0 M2 -1.0 S2 -1.0 N2

2SM2 2 2.0 S2 -1.0 M2

ST38 3 2.0 M2 1.0 S2 -2.0 N2

SKM2 3 1.0 S2 1.0 K2 -1.0 M2

2SN2 2 2.0 S2 -1.0 N2

NO3 2 1.0 N2 1.0 O1

MO3 2 1.0 M2 1.0 O1

NK3 2 1.0 N2 1.0 K1

SO3 2 1.0 S2 1.0 O1

MK3 2 1.0 M2 1.0 K1

SP3 2 1.0 S2 1.0 P1

SK3 2 1.0 S2 1.0 K1

ST8 3 2.0 M2 1.0 N2 -1.0 S2

N4 1 2.0 N2

3MS4 2 3.0 M2 -1.0 S2

ST39 4 1.0 M2 1.0 S2 1.0 N2 -1.0 K2

MN4 2 1.0 M2 1.0 N2

ST40 3 2.0 M2 1.0 S2 -1.0 K2

ST9 4 1.0 M2 1.0 N2 1.0 K2 -1.0 S2

M4 1 2.0 M2

ST10 3 2.0 M2 1.0 K2 -1.0 S2

SN4 2 1.0 S2 1.0 N2

KN4 2 1.0 K2 1.0 N2

MS4 2 1.0 M2 1.0 S2

MK4 2 1.0 M2 1.0 K2

SL4 2 1.0 S2 1.0 L2

S4 1 2.0 S2

SK4 2 1.0 S2 1.0 K2

MNO5 3 1.0 M2 1.0 N2 1.0 O1

2MO5 2 2.0 M2 1.0 O1

3MP5 2 3.0 M2 -1.0 P1

MNK5 3 1.0 M2 1.0 N2 1.0 K1

2MP5 2 2.0 M2 1.0 P1

2MK5 2 2.0 M2 1.0 K1

MSK5 3 1.0 M2 1.0 S2 1.0 K1

3KM5 3 1.0 K2 1.0 K1 1.0 M2

2SK5 2 2.0 S2 1.0 K1

ST11 3 3.0 N2 1.0 K2 -1.0 S2

2NM6 2 2.0 N2 1.0 M2

ST12 4 2.0 N2 1.0 M2 1.0 K2 -1.0 S2

ST41 3 3.0 M2 1.0 S2 -1.0 K2

2MN6 2 2.0 M2 1.0 N2

ST13 4 2.0 M2 1.0 N2 1.0 K2 -1.0 S2

M6 1 3.0 M2

MSN6 3 1.0 M2 1.0 S2 1.0 N2

MKN6 3 1.0 M2 1.0 K2 1.0 N2

2MS6 2 2.0 M2 1.0 S2

2MK6 2 2.0 M2 1.0 K2

NSK6 3 1.0 N2 1.0 S2 1.0 K2

2SM6 2 2.0 S2 1.0 M2

MSK6 3 1.0 M2 1.0 S2 1.0 K2

ST42 3 2.0 M2 2.0 S2 -1.0 K2

S6 1 3.0 S2

ST14 3 2.0 M2 1.0 N2 1.0 O1

ST15 3 2.0 N2 1.0 M2 1.0 K1

M7 1 3.5 M2

46

ST16 3 2.0 M2 1.0 S2 1.0 O1

3MK7 2 3.0 M2 1.0 K1

ST17 4 1.0 M2 1.0 S2 1.0 K2 1.0 O1

ST18 2 2.0 M2 2.0 N2

3MN8 2 3.0 M2 1.0 N2

ST19 4 3.0 M2 1.0 N2 1.0 K2 -1.0 S2

M8 1 4.0 M2

ST20 3 2.0 M2 1.0 S2 1.0 N2

ST21 3 2.0 M2 1.0 N2 1.0 K2

3MS8 2 3.0 M2 1.0 S2

3MK8 2 3.0 M2 1.0 K2

ST22 4 1.0 M2 1.0 S2 1.0 N2 1.0 K2

ST23 2 2.0 M2 2.0 S2

ST24 3 2.0 M2 1.0 S2 1.0 K2

ST25 3 2.0 M2 2.0 N2 1.0 K1

ST26 3 3.0 M2 1.0 N2 1.0 K1

4MK9 2 4.0 M2 1.0 K1

ST27 3 3.0 M2 1.0 S2 1.0 K1

ST28 2 4.0 M2 1.0 N2

M10 1 5.0 M2

ST29 3 3.0 M2 1.0 N2 1.0 S2

ST30 2 4.0 M2 1.0 S2

ST31 4 2.0 M2 1.0 N2 1.0 S2 1.0 K2

ST32 2 3.0 M2 2.0 S2

ST33 3 4.0 M2 1.0 S2 1.0 K1

M12 1 6.0 M2

ST34 2 5.0 M2 1.0 S2

ST35 4 3.0 M2 1.0 N2 1.0 K2 1.0 S2

47

Appendix 7.2 Sample Tidal Station Input Data for the Analysis Program.

The following sample input for logical unit 4 will produce an analysis of Tuktoyaktuk, NorthwestTerritories data for the period 1600 MST July 6, 1975 to 1400 MST September 9, 1975 inclusive,with constituents P1 and K2 inferred, shallow water constituent M10 specifically designated foranalysis inclusion and only line printer output of the results. The final analysis results are listedin Appendix 7.3.

6 1.0 0.0

K1 0.0417807462 P1 0.0415525871 0.33093 -7.07

S2 0.0833333333 K2 0.0835614924 0.27215 -22.40

M10 M8

8 16060775 14090975

6485 TUKTOYUKTUK NWT MST 6927 3302

1 6485 6 775

2 6485 6 775 215 224 215 202 215 227 234 242 238

1 6485 7 775 229 218 206 200 193 187 179 176 183 199 215 231

2 6485 7 775 252 263 260 244 210 176 154 145 153 162 182 203

1 6485 8 775 221 232 230 195 153 119 105 115 132 159 192 218

2 6485 8 775 246 262 264 252 228 197 166 154 159 178 201 229

1 6485 9 775 251 267 291 257 225 204 183 176 188 183 204 232

2 6485 9 775 255 272 285 298 296 253 199 152 112 111 132 167

1 6485 10 775 201 221 227 223 201 166 131 99 70 82 121 161

2 6485 10 775 209 264 302 321 329 303 254 205 168 148 163 180

1 6485 11 775 212 244 271 282 278 258 221 169 135 126 134 156

2 6485 11 775 182 219 249 257 262 243 205 168 135 110 105 118

1 6485 12 775 142 178 213 242 247 233 203 159 119 85 72 89

2 6485 12 775 116 148 180 205 223 222 186 147 105 66 43 55

1 6485 13 775 78 104 136 167 194 199 182 148 107 72 54 61

2 6485 13 775 87 108 139 165 182 190 185 158 125 89 59 49

1 6485 14 775 55 84 113 138 164 182 194 184 154 118 83 66

2 6485 14 775 66 87 117 146 165 180 181 164 134 100 62 40

1 6485 15 775 36 48 72 103 129 157 167 167 156 131 105 87

2 6485 15 775 71 72 80 95 114 128 137 144 132 110 83 63

1 6485 16 775 42 24 29 57 94 125 147 156 155 134 105 82

2 6485 16 775 70 64 63 74 95 116 131 135 130 115 93 75

1 6485 17 775 60 50 50 61 84 111 135 149 154 154 143 122

2 6485 17 775 99 81 72 73 85 99 115 127 127 115 108 94

1 6485 18 775 78 59 50 50 64 81 100 132 164 184 188 179

2 6485 18 775 159 137 133 135 137 143 147 148 153 161 170 161

1 6485 19 775 147 141 142 128 130 143 160 177 193 211 230 236

2 6485 19 775 226 204 181 165 157 161 178 178 178 186 196 198

1 6485 20 775 201 190 172 155 138 130 136 156 174 199 227 245

2 6485 20 775 254 256 245 199 162 141 129 134 157 183 206 220

1 6485 21 775 219 222 210 194 182 169 171 183 206 240 255 265

2 6485 21 775 288 296 292 282 262 238 212 190 186 198 220 240

1 6485 22 775 259 268 271 264 249 228 203 184 187 206 230 257

2 6485 22 775 283 293 295 282 261 232 204 182 165 171 192 218

1 6485 23 775 232 247 255 249 230 205 181 158 148 152 180 209

2 6485 23 775 234 260 272 261 231 196 160 130 111 109 125 157

1 6485 24 775 187 209 224 231 209 181 155 125 110 111 130 159

2 6485 24 775 195 227 249 250 233 200 161 123 94 87 97 123

48

1 6485 25 775 153 183 196 202 195 174 138 101 71 58 60 87

2 6485 25 775 122 159 185 202 199 179 144 103 66 40 35 48

1 6485 26 775 75 104 132 151 160 155 129 98 66 39 34 47

2 6485 26 775 79 113 144 163 172 167 151 117 85 50 20 19

1 6485 27 775 39 74 107 136 148 158 141 118 89 54 29 16

2 6485 27 775 41 76 105 143 189 202 196 185 185 162 160 163

1 6485 28 775 168 187 222 254 260 275 281 268 256 241 221 198

2 6485 28 775 208 230 258 264 285 301 291 270 247 212 188 176

1 6485 29 775 183 200 224 245 256 269 280 270 243 216 194 164

2 6485 29 775 163 177 201 232 263 282 281 290 259 238 202 179

1 6485 30 775 179 184 205 226 242 272 281 279 263 233 205 279

2 6485 30 775 168 184 210 235 247 253 263 259 244 221 193 183

1 6485 31 775 180 176 194 208 215 224 235 243 241 225 207 188

2 6485 31 775 176

1 6485 1 875

2 6485 1 875

1 6485 2 875

2 6485 2 875 104 95 93 95 103 112 118 118 116 108

1 6485 3 875 97 83 68 56 51 54 75 95 117 130 138 139

2 6485 3 875 133 120 103 87 71 56 52 66 81 98 109 107

1 6485 4 875 98 77 49 28 14 4 7 17 44 70 94 110

2 6485 4 875 117 116 107 88 71 55 46 44 60 84 108 125

1 6485 5 875 133 136 114 86 70 62 62 79 113 143 175 208

2 6485 5 875 238 256 266 240 203 179 143 117 118 146 167 186

1 6485 6 875 224 243 227 204 180 158 154 170 201 222 234 243

2 6485 6 875 254 260 247 231 211 188 160 143 137 145 167 195

1 6485 7 875 221 239 249 249 227 184 144 111 102 129 170 201

2 6485 7 875 233 255 260 252 227 195 156 123 107 118 149 180

1 6485 8 875 211 232 245 257 229 200 171 138 102 95 122 163

2 6485 8 875 207 253 295 338 369 353 318 285 221 184 165 175

1 6485 9 875 212 240 260 283 282 259 229 196 174 176 187 204

2 6485 9 875 244 288 329 356 369 370 324 281 289 294 293 287

1 6485 10 875 329 380 426 441 447 453 418 387 353 337 322 314

2 6485 10 875 342 365 404 438 470 482 487 456 441 423 438 448

1 6485 11 875 464 478 491 505 538 528 493 488 472 425 398 390

2 6485 11 875 393 408 421 438 444 433 412 379 337 300 262 247

1 6485 12 875 245 252 277 304 327 339 339 308 257 208 182 182

2 6485 12 875 203 235 260 281 319 315 297 273 237 198 168 158

1 6485 13 875 157 171 195 217 239 252 258 253 242 225 202 179

2 6485 13 875 167 172 190 217 242 257 266 263 244 217 187 155

1 6485 14 875 132 134 163 195 228 246 259 256 236 209 180 150

2 6485 14 875 129 122 136 161 184 200 207 205 195 177 158 136

1 6485 15 875 116 105 104 115 140 164 193 203 216 208 196 187

2 6485 15 875 159 142 147 164 175 183 197 202 202 202 192 176

1 6485 16 875 160 147 137 136 152 172 195 211 224 228 222 210

2 6485 16 875 199 186 171 165 163 169 180 190 201 203 200 193

1 6485 17 875 185 175 162 152 156 169 201 227 249 272 284 285

2 6485 17 875 295 280 259 241 225 211 211 226 247 268 286 297

1 6485 18 875 296 272 245 214 196 194 209 226 239 244 245 248

2 6485 18 875 246 239 229 218 201 183 165 158 160 183 207 221

1 6485 19 875 227 224 209 187 159 138 131 139 162 185 209 228

2 6485 19 875 239 242 233 212 183 152 129 119 132 167 193 218

1 6485 20 875 237 241 230 205 178 151 130 114 122 145 172 203

2 6485 20 875 226 237 237 223 197 165 131 108 103 118 144 173

1 6485 21 875 203 225 229 223 200 175 150 129 131 146 173 202

2 6485 21 875 236 258 263 256 233 198 165 137 127 133 159 190

1 6485 22 875 221 241 252 252 231 200 167 137 119 114 134 166

2 6485 22 875 201 234 256 264 249 212 176 140 111 103 115 140

49

1 6485 23 875 171 203 232 244 242 214 180 146 121 115 126 157

2 6485 23 875 187 211 235 249 247 229 194 151 114 88 87 110

1 6485 24 875 143 177 206 230 237 223 186 140 93 64 66 94

2 6485 24 875 129 165 197 215 220 205 177 144 112 84 80 100

1 6485 25 875 136 173 208 238 252 244 217 181 139 105 89 93

2 6485 25 875 121 159 188 199 200 185 155 121 84 64 45 28

1 6485 26 875 32 72 121 174 215 237 211 197 234 243 176 196

2 6485 26 875 250 219 272 361 391 376 355 389 370 321 300 285

1 6485 27 875 288 323 350 380 422 415 405 389 412 430 453 509

2 6485 27 875 557 559 548 560 576 557 513 489 462 422 388 383

1 6485 28 875 371 393 413 419 443 472 444 423 384 340 304 284

2 6485 28 875 280 286 300 309 312 324 319 299 270 227 203 181

1 6485 29 875 193 240 281 317 352 351 361 350 354 348 358 350

2 6485 29 875 326 317 316 324 327 313 298 283 264 244 230 215

1 6485 30 875 194 194 217 241 256 262 261 259 247 229 213 195

2 6485 30 875 175 163 168 177 187 198 203 204 191 171 144 119

1 6485 31 875 97 92 102 125 150 168 176 188 197 197 206 202

2 6485 31 875 191 186 192 200 197 199 206 205 207 208 205 198

1 6485 1 975 185 187 194 209 234 255 275 285 305 327 332 320

2 6485 1 975 301 295 291 275 277 294 312 328 344 335 321 328

1 6485 2 975 323 315 324 316 318 329 321 314 317 329 336 336

2 6485 2 975 327 316 301 284 263 245 236 231 233 240 250 262

1 6485 3 975 261 250 227 202 172 153 153 162 171 172 190 214

2 6485 3 975 226 228 214 186 160 143 142 155 173 201 236 255

1 6485 4 975 274 284 282 255 216 183 165 179 203 231 258 294

2 6485 4 975 327 364 353 332 299 262 227 207 219 240 256 275

1 6485 5 975 302 309 298 274 240 196 159 142 158 192 222 249

2 6485 5 975 270 280 282 269 239 197 154 110 99 125 159 187

1 6485 6 975 214 235 236 221 189 153 118 83 63 52 65 106

2 6485 6 975 132 151 165 175 169 148 115 74 42 18 5 30

1 6485 7 975 68 123 189 218 198 167 126 81 52 40 56 81

2 6485 7 975 121 173 203 211 199 173 137 100 72 58 57 92

1 6485 8 975 135 179 218 238 240 224 190 136 91 58 48 65

2 6485 8 975 99 140 173 194 195 175 141 96 57 33 30 50

1 6485 9 975 86 129 173 204 217 202 171 125 79 47 40 59

2 6485 9 975 88 121

50

Appendix 7.3 Final Analysis Results Arising from the Input Data of Appendix 7.2and the Standard Constituent Data Package of Appendix 7.1.

ANALYSIS OF HOURLY TIDAL HEIGHTS STN 6485 16H 6/ 7/75 TO 14H 9/ 9/75

NO.OBS.= 1559 NO.PTS.ANAL.= 1559 MIDPT= 3H 8/ 8/75 SEPARATION =1.00

NO NAME FREQUENCY STN M-Y/ M-Y A G AL GL

1 Z0 0.00000000 6485 775/ 975 1.9806 0.00 1.9806 0.00

2 MM 0.00151215 6485 775/ 975 0.2121 263.34 0.2121 288.50

3 MSF 0.00282193 6485 775/ 975 0.1561 133.80 0.1561 115.15

4 ALP1 0.03439657 6485 775/ 975 0.0152 334.95 0.0141 180.96

5 2Q1 0.03570635 6485 775/ 975 0.0246 82.69 0.0226 246.82

6 Q1 0.03721850 6485 775/ 975 0.0158 65.74 0.0144 252.75

7 O1 0.03873065 6485 775/ 975 0.0764 74.23 0.0694 284.43

8 NO1 0.04026859 6485 775/ 975 0.0290 238.14 0.0380 275.85

9 P1 0.04155259 6485 775/ 975 0.0465 71.76 INF FR K1 0.0468 252.20

10 K1 0.04178075 6485 775/ 975 0.1406 64.69 0.1332 145.54

11 J1 0.04329290 6485 775/ 975 0.0253 7.32 0.0234 103.63

12 OO1 0.04483084 6485 775/ 975 0.0531 235.74 0.0463 358.47

13 UPS1 0.04634299 6485 775/ 975 0.0298 91.73 0.0233 239.12

14 EPS2 0.07617731 6485 775/ 975 0.0211 184.59 0.0216 109.98

15 MU2 0.07768947 6485 775/ 975 0.0419 83.23 0.0428 30.06

16 N2 0.07899925 6485 775/ 975 0.0838 44.52 0.0857 306.35

17 M2 0.08051140 6485 775/ 975 0.4904 77.70 0.5007 4.40

18 L2 0.08202355 6485 775/ 975 0.0213 35.21 0.0174 168.03

19 S2 0.08333334 6485 775/ 975 0.2195 126.65 0.2193 36.74

20 K2 0.08356149 6485 775/ 975 0.0597 149.05 INF FR S2 0.0515 131.15

21 ETA2 0.08507364 6485 775/ 975 0.0071 246.05 0.0059 235.38

22 MO3 0.11924206 6485 775/ 975 0.0148 234.97 0.0138 11.86

23 M3 0.12076710 6485 775/ 975 0.0123 261.57 0.0126 331.91

24 MK3 0.12229215 6485 775/ 975 0.0049 331.60 0.0048 339.15

25 SK3 0.12511408 6485 775/ 975 0.0023 237.69 0.0022 228.64

26 MN4 0.15951066 6485 775/ 975 0.0092 256.47 0.0096 85.00

27 M4 0.16102280 6485 775/ 975 0.0126 291.78 0.0131 145.17

28 SN4 0.16233259 6485 775/ 975 0.0083 270.85 0.0085 82.78

29 MS4 0.16384473 6485 775/ 975 0.0010 339.35 0.0011 176.14

30 S4 0.16666667 6485 775/ 975 0.0047 299.56 0.0047 119.75

31 2MK5 0.20280355 6485 775/ 975 0.0013 310.10 0.0013 244.34

32 2SK5 0.20844743 6485 775/ 975 0.0045 104.00 0.0043 5.04

33 2MN6 0.24002205 6485 775/ 975 0.0035 271.24 0.0038 26.46

34 M6 0.24153420 6485 775/ 975 0.0017 158.89 0.0018 298.97

35 2MS6 0.24435614 6485 775/ 975 0.0056 306.10 0.0059 69.59

36 2SM6 0.24717808 6485 775/ 975 0.0023 298.92 0.0023 45.80

37 3MK7 0.28331494 6485 775/ 975 0.0086 212.25 0.0086 73.20

38 M8 0.32204559 6485 775/ 975 0.0030 42.43 0.0033 109.22

39 M10 0.40255699 6485 775/ 975 0.0009 198.23 0.0010 191.71

51

Appendix 7.4 Sample Input for the Tidal Heights Prediction Program.

The following sample input for logical unit 8 will synthesize hourly heights and the times andheights of all extrema at Victoria, British Columbia for the period 0100 PST July 1, 1976 to2400 PST July 31, 1976 inclusive. The output results are listed in Appendix 7.5.

.7428797055 .7771900329 .5187051308 .3631582592 .7847990160 000GMT 1/1/76

13.3594019864 .9993368945 .1129517942 .0536893056 .0000477414 INCR./365DAYS

Z0 0 0 0 0 0 0 0.0 0

SA 0 0 1 0 0 -1 0.0 0

SSA 0 0 2 0 0 0 0.0 0

MSM 0 1 -2 1 0 0 .00 0

MM 0 1 0 -1 0 0 0.0 0

MSF 0 2 -2 0 0 0 0.0 0

MF 0 2 0 0 0 0 0.0 0

ALP1 1 -4 2 1 0 0 -.25 2

ALP1 -1 0 0 .75 0.0360R1 0 -1 0 .00 0.1906

2Q1 1 -3 0 2 0 0-0.25 5

2Q1 -2 -2 0 .50 0.0063 -1 -1 0 .75 0.0241R1 -1 0 0 .75 0.0607R1

2Q1 0 -2 0 .50 0.0063 0 -1 0 .0 0.1885

SIG1 1 -3 2 0 0 0-0.25 4

SIG1 -1 0 0 .75 0.0095R1 0 -2 0 .50 0.0061 0 -1 0 .0 0.1884

SIG1 2 0 0 .50 0.0087

Q1 1 -2 0 1 0 0-0.25 10

Q1 -2 -3 0 .50 0.0007 -2 -2 0 .50 0.0039 -1 -2 0 .75 0.0010R1

Q1 -1 -1 0 .75 0.0115R1 -1 0 0 .75 0.0292R1 0 -2 0 .50 0.0057

Q1 -1 0 1 .0 0.0008 0 -1 0 .0 0.1884 1 0 0 .75 0.0018R1

Q1 2 0 0 .50 0.0028

RHO1 1 -2 2 -1 0 0-0.25 5

RHO1 0 -2 0 .50 0.0058 0 -1 0 .0 0.1882 1 0 0 .75 0.0131R1

RHO1 2 0 0 .50 0.0576 2 1 0 .0 0.0175

O1 1 -1 0 0 0 0-0.25 8

O1 -1 0 0 .25 0.0003R1 0 -2 0 .50 0.0058 0 -1 0 .0 0.1885

O1 1 -1 0 .25 0.0004R1 1 0 0 .75 0.0029R1 1 1 0 .25 0.0004R1

O1 2 0 0 .50 0.0064 2 1 0 .50 0.0010

TAU1 1 -1 2 0 0 0-0.75 5

TAU1 -2 0 0 .0 0.0446 -1 0 0 .25 0.0426R1 0 -1 0 .50 0.0284

TAU1 0 1 0 .50 0.2170 0 2 0 .50 0.0142

BET1 1 0 -2 1 0 0 -.75 1

BET1 0 -1 0 .00 0.2266

NO1 1 0 0 1 0 0-0.75 9

NO1 -2 -2 0 .50 0.0057 -2 -1 0 .0 0.0665 -2 0 0 .0 0.3596

NO1 -1 -1 0 .75 0.0331R1 -1 0 0 .25 0.2227R1 -1 1 0 .75 0.0290R1

NO1 0 -1 0 .50 0.0290 0 1 0 .0 0.2004 0 2 0 .50 0.0054

CHI1 1 0 2 -1 0 0-0.75 2

CHI1 0 -1 0 .50 0.0282 0 1 0 .0 0.2187

PI1 1 1 -3 0 0 1-0.25 1

PI1 0 -1 0 .50 0.0078

P1 1 1 -2 0 0 0-0.25 6

P1 0 -2 0 .0 0.0008 0 -1 0 .50 0.0112 0 0 2 .50 0.0004

P1 1 0 0 .75 0.0004R1 2 0 0 .50 0.0015 2 1 0 .50 0.0003

S1 1 1 -1 0 0 1-0.75 2

S1 0 0 -2 .0 0.3534 0 1 0 .50 0.0264

52

K1 1 1 0 0 0 0-0.75 10

K1 -2 -1 0 .0 0.0002 -1 -1 0 .75 0.0001R1 -1 0 0 .25 0.0007R1

K1 -1 1 0 .75 0.0001R1 0 -2 0 .0 0.0001 0 -1 0 .50 0.0198

K1 0 1 0 .0 0.1356 0 2 0 .50 0.0029 1 0 0 .25 0.0002R1

K1 1 1 0 .25 0.0001R1

PSI1 1 1 1 0 0 -1-0.75 1

PSI1 0 1 0 .0 0.0190

PHI1 1 1 2 0 0 0-0.75 5

PHI1 -2 0 0 .0 0.0344 -2 1 0 .0 0.0106 0 0 -2 .0 0.0132

PHI1 0 1 0 .50 0.0384 0 2 0 .50 0.0185

THE1 1 2 -2 1 0 0 -.75 4

THE1 -2 -1 0 .00 .0300 -1 0 0 .25 0.0141R1 0 -1 0 .50 .0317

THE1 0 1 0 .00 .1993

J1 1 2 0 -1 0 0-0.75 10

J1 0 -1 0 .50 0.0294 0 1 0 .0 0.1980 0 2 0 .50 0.0047

J1 1 -1 0 .75 0.0027R1 1 0 0 .25 0.0816R1 1 1 0 .25 0.0331R1

J1 1 2 0 .25 0.0027R1 2 0 0 .50 0.0152 2 1 0 .50 0.0098

J1 2 2 0 .50 0.0057

OO1 1 3 0 0 0 0-0.75 8

OO1 -2 -1 0 .50 0.0037 -2 0 0 .0 0.1496 -2 1 0 .0 0.0296

OO1 -1 0 0 .25 0.0240R1 -1 1 0 .25 0.0099R1 0 1 0 .0 0.6398

OO1 0 2 0 .0 0.1342 0 3 0 .0 0.0086

UPS1 1 4 0 -1 0 0 -.75 5

UPS1 -2 0 0 .00 0.0611 0 1 0 .00 0.6399 0 2 0 .00 0.1318

UPS1 1 0 0 .25 0.0289R1 1 1 0 .25 0.0257R1

OQ2 2 -3 0 3 0 0 0.0 2

OQ2 -1 0 0 .25 0.1042R2 0 -1 0 .50 0.0386

EPS2 2 -3 2 1 0 0 0.0 3

EPS2 -1 -1 0 .25 0.0075R2 -1 0 0 .25 0.0402R2 0 -1 0 .50 0.0373

2N2 2 -2 0 2 0 0 0.0 4

2N2 -2 -2 0 .50 0.0061 -1 -1 0 .25 0.0117R2 -1 0 0 .25 0.0678R2

2N2 0 -1 0 .50 0.0374

MU2 2 -2 2 0 0 0 0.0 3

MU2 -1 -1 0 .25 0.0018R2 -1 0 0 .25 0.0104R2 0 -1 0 .50 0.0375

N2 2 -1 0 1 0 0 0.0 4

N2 -2 -2 0 .50 0.0039 -1 0 1 .00 0.0008 0 -2 0 .00 0.0005

N2 0 -1 0 .50 0.0373

NU2 2 -1 2 -1 0 0 0.0 4

NU2 0 -1 0 .50 0.0373 1 0 0 .75 0.0042R2 2 0 0 .0 0.0042

NU2 2 1 0 .50 0.0036

GAM2 2 0 -2 2 0 0 -.50 3

GAM2 -2 -2 0 .00 0.1429 -1 0 0 .25 0.0293R2 0 -1 0 .50 0.0330

H1 2 0 -1 0 0 1-0.50 2

H1 0 -1 0 .50 0.0224 1 0 -1 .50 0.0447

M2 2 0 0 0 0 0 0.0 9

M2 -1 -1 0 .75 0.0001R2 -1 0 0 .75 0.0004R2 0 -2 0 .0 0.0005

M2 0 -1 0 .50 0.0373 1 -1 0 .25 0.0001R2 1 0 0 .75 0.0009R2

M2 1 1 0 .75 0.0002R2 2 0 0 .0 0.0006 2 1 0 .0 0.0002

H2 2 0 1 0 0 -1 0.0 1

H2 0 -1 0 .50 0.0217

LDA2 2 1 -2 1 0 0-0.50 1

LDA2 0 -1 0 .50 0.0448

L2 2 1 0 -1 0 0-0.50 5

L2 0 -1 0 .50 0.0366 2 -1 0 .00 0.0047 2 0 0 .50 0.2505

L2 2 1 0 .50 0.1102 2 2 0 .50 0.0156

T2 2 2 -3 0 0 1 0.0 0

S2 2 2 -2 0 0 0 0.0 3

S2 0 -1 0 .0 0.0022 1 0 0 .75 0.0001R2 2 0 0 .0 0.0001

53

R2 2 2 -1 0 0 -1-0.50 2

R2 0 0 2 .50 0.2535 0 1 2 .0 0.0141

K2 2 2 0 0 0 0 0.0 5

K2 -1 0 0 .75 0.0024R2 -1 1 0 .75 0.0004R2 0 -1 0 .50 0.0128

K2 0 1 0 .0 0.2980 0 2 0 .0 0.0324

ETA2 2 3 0 -1 0 0 0.0 7

ETA2 0 -1 0 .50 0.0187 0 1 0 .0 0.4355 0 2 0 .0 0.0467

ETA2 1 0 0 .75 0.0747R2 1 1 0 .75 0.0482R2 1 2 0 .75 0.0093R2

ETA2 2 0 0 .50 0.0078

M3 3 0 0 0 0 0 -.50 1

M3 0 -1 0 .50 .0564

2PO1 2 2.0 P1 -1.0 O1

SO1 2 1.0 S2 -1.0 O1

ST36 3 2.0 M2 1.0 N2 -2.0 S2

2NS2 2 2.0 N2 -1.0 S2

ST37 2 3.0 M2 -2.0 S2

ST1 3 2.0 N2 1.0 K2 -2.0 S2

ST2 4 1.0 M2 1.0 N2 1.0 K2 -2.0 S2

ST3 3 2.0 M2 1.0 S2 -2.0 K2

O2 1 2.0 O1

ST4 3 2.0 K2 1.0 N2 -2.0 S2

SNK2 3 1.0 S2 1.0 N2 -1.0 K2

OP2 2 1.0 O1 1.0 P1

MKS2 3 1.0 M2 1.0 K2 -1.0 S2

ST5 3 1.0 M2 2.0 K2 -2.0 S2

ST6 4 2.0 S2 1.0 N2 -1.0 M2 -1.0 K2

2SK2 2 2.0 S2 -1.0 K2

MSN2 3 1.0 M2 1.0 S2 -1.0 N2

ST7 4 2.0 K2 1.0 M2 -1.0 S2 -1.0 N2

2SM2 2 2.0 S2 -1.0 M2

ST38 3 2.0 M2 1.0 S2 -2.0 N2

SKM2 3 1.0 S2 1.0 K2 -1.0 M2

2SN2 2 2.0 S2 -1.0 N2

NO3 2 1.0 N2 1.0 O1

MO3 2 1.0 M2 1.0 O1

NK3 2 1.0 N2 1.0 K1

SO3 2 1.0 S2 1.0 O1

MK3 2 1.0 M2 1.0 K1

SP3 2 1.0 S2 1.0 P1

SK3 2 1.0 S2 1.0 K1

ST8 3 2.0 M2 1.0 N2 -1.0 S2

N4 1 2.0 N2

3MS4 2 3.0 M2 -1.0 S2

ST39 4 1.0 M2 1.0 S2 1.0 N2 -1.0 K2

MN4 2 1.0 M2 1.0 N2

ST40 3 2.0 M2 1.0 S2 -1.0 K2

ST9 4 1.0 M2 1.0 N2 1.0 K2 -1.0 S2

M4 1 2.0 M2

ST10 3 2.0 M2 1.0 K2 -1.0 S2

SN4 2 1.0 S2 1.0 N2

KN4 2 1.0 K2 1.0 N2

MS4 2 1.0 M2 1.0 S2

MK4 2 1.0 M2 1.0 K2

SL4 2 1.0 S2 1.0 L2

S4 1 2.0 S2

SK4 2 1.0 S2 1.0 K2

54

MNO5 3 1.0 M2 1.0 N2 1.0 O1

2MO5 2 2.0 M2 1.0 O1

3MP5 2 3.0 M2 -1.0 P1

MNK5 3 1.0 M2 1.0 N2 1.0 K1

2MP5 2 2.0 M2 1.0 P1

2MK5 2 2.0 M2 1.0 K1

MSK5 3 1.0 M2 1.0 S2 1.0 K1

3KM5 3 1.0 K2 1.0 K1 1.0 M2

2SK5 2 2.0 S2 1.0 K1

ST11 3 3.0 N2 1.0 K2 -1.0 S2

2NM6 2 2.0 N2 1.0 M2

ST12 4 2.0 N2 1.0 M2 1.0 K2 -1.0 S2

ST41 3 3.0 M2 1.0 S2 -1.0 K2

2MN6 2 2.0 M2 1.0 N2

ST13 4 2.0 M2 1.0 N2 1.0 K2 -1.0 S2

M6 1 3.0 M2

MSN6 3 1.0 M2 1.0 S2 1.0 N2

MKN6 3 1.0 M2 1.0 K2 1.0 N2

2MS6 2 2.0 M2 1.0 S2

2MK6 2 2.0 M2 1.0 K2

NSK6 3 1.0 N2 1.0 S2 1.0 K2

2SM6 2 2.0 S2 1.0 M2

MSK6 3 1.0 M2 1.0 S2 1.0 K2

ST42 3 2.0 M2 2.0 S2 -1.0 K2

S6 1 3.0 S2

ST14 3 2.0 M2 1.0 N2 1.0 O1

ST15 3 2.0 N2 1.0 M2 1.0 K1

M7 1 3.5 M2

ST16 3 2.0 M2 1.0 S2 1.0 O1

3MK7 2 3.0 M2 1.0 K1

ST17 4 1.0 M2 1.0 S2 1.0 K2 1.0 O1

ST18 2 2.0 M2 2.0 N2

3MN8 2 3.0 M2 1.0 N2

ST19 4 3.0 M2 1.0 N2 1.0 K2 -1.0 S2

M8 1 4.0 M2

ST20 3 2.0 M2 1.0 S2 1.0 N2

ST21 3 2.0 M2 1.0 N2 1.0 K2

3MS8 2 3.0 M2 1.0 S2

3MK8 2 3.0 M2 1.0 K2

ST22 4 1.0 M2 1.0 S2 1.0 N2 1.0 K2

ST23 2 2.0 M2 2.0 S2

ST24 3 2.0 M2 1.0 S2 1.0 K2

ST25 3 2.0 M2 2.0 N2 1.0 K1

ST26 3 3.0 M2 1.0 N2 1.0 K1

4MK9 2 4.0 M2 1.0 K1

ST27 3 3.0 M2 1.0 S2 1.0 K1

ST28 2 4.0 M2 1.0 N2

M10 1 5.0 M2

ST29 3 3.0 M2 1.0 N2 1.0 S2

ST30 2 4.0 M2 1.0 S2

ST31 4 2.0 M2 1.0 N2 1.0 S2 1.0 K2

ST32 2 3.0 M2 2.0 S2

ST33 3 4.0 M2 1.0 S2 1.0 K1

M12 1 6.0 M2

ST34 2 5.0 M2 1.0 S2

ST35 4 3.0 M2 1.0 N2 1.0 K2 1.0 S2

55

7120 VICTORIA HARBOUR BC PST 48 23 123 22

Z0 6.0670 .00

Q1 .1970 130.30

O1 1.2110 137.00

NO1 0.1120 120.80

P1 .6740 148.50

S1 .0980 154.10

K1 2.0700 149.40

J1 .1170 166.40

N2 .2940 63.40

M2 1.2130 87.00

S2 .3320 93.90

001007076 031007076 EQUI 1.0

001007076 031007076 EXTR 0.5

56

Appendix 7.5 Tidal Heights Prediction Results Arising from the Input Data of Appendix 7.4.Figure 2 is the Plot of These Hourly Heights over the Period 0100 PST July 24, 1976

to 2400 PST July 31, 1976.

STN 1ST HR DATE 1 2 3 4 5 6 7 8 DT HRS

7120 1.0000 1 776 7.459 7.736 7.926 7.886 7.518 6.799 5.797 4.658 1.0000

7120 9.0000 1 776 3.578 2.759 2.361 2.470 3.068 4.047 5.226 6.393 1.0000

7120 17.0000 1 776 7.356 7.979 8.211 8.092 7.732 7.283 6.896 6.676 1.0000

7120 1.0000 2 776 6.664 6.823 7.051 7.216 7.189 6.887 6.296 5.482 1.0000

7120 9.0000 2 776 4.581 3.766 3.210 3.044 3.323 4.012 4.995 6.097 1.0000

7120 17.0000 2 776 7.122 7.899 8.314 8.338 8.022 7.484 6.878 6.351 1.0000

7120 1.0000 3 776 6.011 5.900 5.988 6.186 6.375 6.436 6.289 5.914 1.0000

7120 9.0000 3 776 5.363 4.747 4.211 3.896 3.911 4.293 5.004 5.934 1.0000

7120 17.0000 3 776 6.919 7.782 8.370 8.586 8.407 7.893 7.164 6.375 1.0000

7120 1.0000 4 776 5.680 5.195 4.976 5.009 5.220 5.496 5.722 5.807 1.0000

7120 9.0000 4 776 5.712 5.462 5.136 4.851 4.727 4.857 5.277 5.957 1.0000

7120 17.0000 4 776 6.797 7.652 8.357 8.770 8.798 8.424 7.706 6.770 1.0000

7120 1.0000 5 776 5.778 4.899 4.267 3.958 3.975 4.255 4.688 5.143 1.0000

7120 9.0000 5 776 5.509 5.718 5.761 5.690 5.598 5.596 5.775 6.182 1.0000

7120 17.0000 5 776 6.798 7.539 8.271 8.836 9.093 8.948 8.381 7.454 1.0000

7120 1.0000 6 776 6.301 5.102 4.046 3.292 2.936 2.996 3.409 4.053 1.0000

7120 9.0000 6 776 4.775 5.432 5.923 6.211 6.325 6.350 6.399 6.574 1.0000

7120 17.0000 6 776 6.935 7.479 8.131 8.762 9.210 9.329 9.020 8.263 1.0000

7120 1.0000 7 776 7.127 5.764 4.376 3.180 2.357 2.019 2.187 2.787 1.0000

7120 9.0000 7 776 3.672 4.661 5.576 6.287 6.737 6.950 7.014 7.053 1.0000

7120 17.0000 7 776 7.188 7.494 7.976 8.561 9.114 9.465 9.458 8.991 1.0000

7120 1.0000 8 776 8.050 6.720 5.176 3.648 2.377 1.560 1.314 1.648 1.0000

7120 9.0000 8 776 2.465 3.589 4.803 5.903 6.743 7.259 7.482 7.516 1.0000

7120 17.0000 8 776 7.507 7.588 7.846 8.286 8.827 9.320 9.584 9.456 1.0000

7120 1.0000 9 776 8.838 7.730 6.241 4.572 2.979 1.719 0.993 0.906 1.0000

7120 9.0000 9 776 1.444 2.478 3.799 5.164 6.352 7.215 7.701 7.860 1.0000

7120 17.0000 9 776 7.821 7.744 7.777 8.005 8.422 8.932 9.371 9.548 1.0000

7120 1.0000 10 776 9.302 8.547 7.306 5.715 4.002 2.437 1.282 0.725 1.0000

7120 9.0000 10 776 0.846 1.596 2.812 4.258 5.675 6.844 7.632 8.010 1.0000

7120 17.0000 10 776 8.057 7.923 7.782 7.782 7.996 8.401 8.881 9.260 1.0000

7120 1.0000 11 776 9.345 8.986 8.121 6.800 5.186 3.522 2.086 1.121 1.0000

7120 9.0000 11 776 0.791 1.138 2.074 3.408 4.886 6.253 7.306 7.941 1.0000

7120 17.0000 11 776 8.163 8.077 7.850 7.660 7.643 7.851 8.240 8.679 1.0000

7120 1.0000 12 776 8.984 8.974 8.519 7.586 6.255 4.707 3.191 1.971 1.0000

7120 9.0000 12 776 1.261 1.184 1.740 2.808 4.176 5.591 6.816 7.681 1.0000

7120 17.0000 12 776 8.116 8.163 7.948 7.646 7.424 7.398 7.597 7.956 1.0000

7120 1.0000 13 776 8.333 8.553 8.452 7.933 6.991 5.729 4.337 3.054 1.0000

7120 9.0000 13 776 2.113 1.688 1.854 2.573 3.699 5.015 6.284 7.301 1.0000

7120 17.0000 13 776 7.935 8.158 8.038 7.714 7.356 7.115 7.078 7.253 1.0000

7120 1.0000 14 776 7.559 7.858 7.990 7.820 7.283 6.400 5.287 4.128 1.0000

7120 9.0000 14 776 3.136 2.502 2.353 2.722 3.537 4.642 5.830 6.894 1.0000

7120 17.0000 14 776 7.667 8.063 8.086 7.823 7.415 7.018 6.758 6.698 1.0000

7120 1.0000 15 776 6.827 7.064 7.279 7.338 7.138 6.641 5.886 4.988 1.0000

7120 9.0000 15 776 4.112 3.432 3.096 3.184 3.691 4.528 5.543 6.550 1.0000

7120 17.0000 15 776 7.379 7.906 8.080 7.931 7.553 7.081 6.649 6.360 1.0000

7120 1.0000 16 776 6.260 6.329 6.490 6.636 6.657 6.479 6.083 5.514 1.0000

7120 9.0000 16 776 4.873 4.295 3.914 3.832 4.092 4.665 5.459 6.333 1.0000

7120 17.0000 16 776 7.135 7.730 8.029 8.012 7.720 7.248 6.718 6.245 1.0000

7120 1.0000 17 776 5.915 5.762 5.769 5.871 5.981 6.016 5.919 5.677 1.0000

7120 9.0000 17 776 5.331 4.958 4.658 4.528 4.634 4.992 5.566 6.269 1.0000

7120 17.0000 17 776 6.981 7.578 7.959 8.062 7.883 7.469 6.910 6.316 1.0000

57

7120 1.0000 18 776 5.790 5.408 5.205 5.168 5.249 5.378 5.486 5.522 1.0000

7120 9.0000 18 776 5.471 5.355 5.226 5.154 5.207 5.426 5.817 6.345 1.0000

7120 17.0000 18 776 6.933 7.484 7.898 8.095 8.031 7.708 7.178 6.525 1.0000

7120 1.0000 19 776 5.855 5.266 4.835 4.602 4.564 4.681 4.892 5.128 1.0000

7120 9.0000 19 776 5.334 5.481 5.571 5.635 5.721 5.880 6.146 6.522 1.0000

7120 17.0000 19 776 6.980 7.456 7.867 8.127 8.167 7.952 7.489 6.833 1.0000

7120 1.0000 20 776 6.071 5.313 4.664 4.206 3.987 4.006 4.225 4.576 1.0000

7120 9.0000 20 776 4.980 5.366 5.689 5.935 6.122 6.291 6.489 6.752 1.0000

7120 17.0000 20 776 7.090 7.479 7.863 8.162 8.295 8.194 7.829 7.215 1.0000

7120 1.0000 21 776 6.415 5.531 4.686 3.997 3.556 3.409 3.554 3.935 1.0000

7120 9.0000 21 776 4.468 5.052 5.599 6.049 6.382 6.616 6.799 6.987 1.0000

7120 17.0000 21 776 7.223 7.524 7.865 8.185 8.399 8.420 8.179 7.652 1.0000

7120 1.0000 22 776 6.868 5.911 4.904 3.992 3.306 2.940 2.935 3.265 1.0000

7120 9.0000 22 776 3.851 4.579 5.324 5.983 6.491 6.835 7.049 7.196 1.0000

7120 17.0000 22 776 7.349 7.560 7.842 8.163 8.447 8.592 8.503 8.112 1.0000

7120 1.0000 23 776 7.407 6.440 5.326 4.217 3.280 2.656 2.433 2.628 1.0000

7120 9.0000 23 776 3.185 3.988 4.890 5.747 6.448 6.937 7.223 7.364 1.0000

7120 17.0000 23 776 7.450 7.567 7.770 8.062 8.390 8.654 8.735 8.528 1.0000

7120 1.0000 24 776 7.973 7.081 5.936 4.689 3.525 2.625 2.130 2.109 1.0000

7120 9.0000 24 776 2.545 3.339 4.336 5.360 6.255 6.917 7.315 7.488 1.0000

7120 17.0000 24 776 7.528 7.549 7.646 7.866 8.193 8.544 8.793 8.803 1.0000

7120 1.0000 25 776 8.469 7.746 6.677 5.386 4.057 2.903 2.111 1.808 1.0000

7120 9.0000 25 776 2.030 2.716 3.722 4.856 5.925 6.775 7.322 7.570 1.0000

7120 17.0000 25 776 7.597 7.529 7.495 7.593 7.854 8.229 8.607 8.835 1.0000

7120 1.0000 26 776 8.766 8.303 7.429 6.220 4.840 3.505 2.439 1.823 1.0000

7120 9.0000 26 776 1.756 2.232 3.142 4.301 5.494 6.520 7.241 7.611 1.0000

7120 17.0000 26 776 7.671 7.541 7.371 7.301 7.418 7.728 8.152 8.547 1.0000

7120 1.0000 27 776 8.744 8.597 8.023 7.038 5.755 4.368 3.113 2.211 1.0000

7120 9.0000 27 776 1.822 2.007 2.713 3.790 5.022 6.182 7.080 7.608 1.0000

7120 17.0000 27 776 7.758 7.616 7.332 7.074 6.982 7.123 7.477 7.934 1.0000

7120 1.0000 28 776 8.333 8.495 8.285 7.643 6.615 5.344 4.043 2.950 1.0000

7120 9.0000 28 776 2.270 2.131 2.551 3.435 4.600 5.815 6.856 7.559 1.0000

7120 17.0000 28 776 7.851 7.766 7.424 6.998 6.664 6.551 6.707 7.085 1.0000

7120 1.0000 29 776 7.556 7.946 8.084 7.847 7.201 6.216 5.052 3.924 1.0000

7120 9.0000 29 776 3.058 2.630 2.731 3.338 4.326 5.492 6.607 7.464 1.0000

7120 17.0000 29 776 7.931 7.972 7.656 7.128 6.573 6.166 6.022 6.169 1.0000

7120 1.0000 30 776 6.545 7.011 7.395 7.535 7.328 6.757 5.905 4.933 1.0000

7120 9.0000 30 776 4.045 3.439 3.258 3.557 4.283 5.293 6.383 7.336 1.0000

7120 17.0000 30 776 7.973 8.192 7.991 7.462 6.766 6.091 5.602 5.405 1.0000

7120 1.0000 31 776 5.516 5.866 6.321 6.716 6.903 6.793 6.379 5.738 1.0000

7120 9.0000 31 776 5.019 4.402 4.051 4.079 4.511 5.281 6.241 7.198 1.0000

7120 17.0000 31 776 7.957 8.368 8.355 7.938 7.223 6.375 5.578 4.992 1.0000

HL STN DATE TIME HGT TIME HGT TIME HGT TIME HGT TIME HGT TIME HGT

0 7120 1 776 322 7.9 1117 2.3 1907 8.2 9999 99.9 9999 99.9 9999 99.9

1 7120 2 776 33 6.6 424 7.2 1153 3.0 1933 8.4 9999 99.9 9999 99.9

1 7120 3 776 200 5.9 550 6.4 1228 3.9 2002 8.6 9999 99.9 9999 99.9

1 7120 4 776 321 5.0 759 5.8 1302 4.7 2034 8.8 9999 99.9 9999 99.9

1 7120 5 776 426 3.9 1047 5.8 1333 5.6 2110 9.1 9999 99.9 9999 99.9

1 7120 6 776 521 2.9 2148 9.3 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 7 776 609 2.0 2230 9.5 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 8 776 655 1.3 1548 7.5 1648 7.5 2313 9.6 9999 99.9 9999 99.9

1 7120 9 776 738 0.9 1611 7.9 1819 7.7 2358 9.5 9999 99.9 9999 99.9

1 7120 10 776 819 0.7 1640 8.1 1931 7.8 9999 99.9 9999 99.9 9999 99.9

0 7120 11 776 44 9.4 859 0.8 1709 8.2 2035 7.6 9999 99.9 9999 99.9

0 7120 12 776 129 9.0 937 1.1 1737 8.2 2137 7.4 9999 99.9 9999 99.9

58

0 7120 13 776 215 8.6 1013 1.7 1806 8.2 2240 7.1 9999 99.9 9999 99.9

0 7120 14 776 300 8.0 1047 2.3 1833 8.1 2347 6.7 9999 99.9 9999 99.9

0 7120 15 776 346 7.3 1118 3.1 1900 8.1 9999 99.9 9999 99.9 9999 99.9

1 7120 16 776 102 6.3 438 6.7 1145 3.8 1926 8.1 9999 99.9 9999 99.9

1 7120 17 776 226 5.7 549 6.0 1205 4.5 1951 8.1 9999 99.9 9999 99.9

1 7120 18 776 345 5.2 755 5.5 1209 5.2 2016 8.1 9999 99.9 9999 99.9

1 7120 19 776 442 4.6 2040 8.2 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 20 776 524 4.0 2106 8.3 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 21 776 559 3.4 2136 8.4 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 22 776 631 2.9 2210 8.6 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 23 776 701 2.4 2250 8.7 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 24 776 732 2.1 2333 8.8 9999 99.9 9999 99.9 9999 99.9 9999 99.9

1 7120 25 776 804 1.8 1639 7.6 1850 7.5 9999 99.9 9999 99.9 9999 99.9

0 7120 26 776 19 8.9 837 1.7 1644 7.7 1955 7.3 9999 99.9 9999 99.9

0 7120 27 776 108 8.8 911 1.8 1657 7.8 2055 7.0 9999 99.9 9999 99.9

0 7120 28 776 159 8.5 945 2.1 1714 7.9 2155 6.6 9999 99.9 9999 99.9

0 7120 29 776 254 8.1 1019 2.6 1736 8.0 2259 6.0 9999 99.9 9999 99.9

0 7120 30 776 356 7.5 1053 3.3 1800 8.2 9999 99.9 9999 99.9 9999 99.9

1 7120 31 776 7 5.4 509 6.9 1126 4.0 1828 8.4 9999 99.9 9999 99.9